Wikipedia:Reference desk/Archives/Mathematics/2011 July 12

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July 12

Simultaneous equations

I would like to know if there is an analytic way to find all the solutions to this pair of simultaneous equations:
$y^{2}-x^{2}+1+2xy=0$
$y^{2}-x^{2}-1-2xy=0$
I can find some of the solutions by inspection: I can find the solutions where $x=y$ or $x=-y$ because then the $y^{2}-x^{2}$ term disappears and the two equations become exactly the same. When $x=y$ we have $2x^{2}+1=0$ which has solutions $x=y={\frac {i}{\sqrt {2}}}$ and $x=y={\frac {-i}{\sqrt {2}}}$. When $x=-y$ we have $2x^{2}-1=0$ which has solutions $x={\frac {1}{\sqrt {2}}},y={\frac {-1}{\sqrt {2}}}$ and $x={\frac {-1}{\sqrt {2}}},y={\frac {1}{\sqrt {2}}}$. Widener (talk) 09:02, 12 July 2011 (UTC)

Subtract one equation from the other and you get

$2+4xy=0$

$\Rightarrow y={\frac {-1}{2x}}$

Substituting this in either equation gives

${\frac {1}{4x^{2}}}-x^{2}=0$

$\Rightarrow x^{4}={\frac {1}{4}}$

$\Rightarrow x=\pm {\frac {1}{\sqrt {2}}}{\text{ , }}\pm {\frac {i}{\sqrt {2}}}$

So the four solutions you have found are the only solutions. Gandalf61 (talk) 09:23, 12 July 2011 (UTC)

Analog of Bezout's Identity with d=2, a,b,x,y odd

I have been generating pairs of consecutive numbers with certain properties (which are preserved under multiplication) by using Bézout's identity; I construct two numbers with partial properties, use Bezout's identity to generate consecutive candidate numbers based on the original two, then check the candidates to see if the extra factors have completed the properties.

I would now like to do a similar thing, but generate pairs of consecutive ODD numbers (differing by 2) with the same properties. I do not see how to adapt Bezout's identity to generate a difference of 2, without introducing a factor of two into every term. A second consideration is to keep the generated factors odd, so that I am generating consecutive odd numbers.

Does anyone have any ideas? Walt (talk) 11:34, 12 July 2011 (UTC)

Could you give us some more details please? You mention Bézout's identity and you mention that d = 2 while a, b, x and y are odd. Bézout's identity tells us that if gcd(a,b) = 2 then there exist integers x and y such that ax + by = 2. If I recall, you can use the Chinese remainder theorem to calculate x and y. Could you please explain a little more about the motivation, and about what these "partial" properties are? An explicit example would be great. — Fly by Night (talk) 00:02, 13 July 2011 (UTC)

Well, it's not really Bezout's identity because 2 is not the GCD of the two odd numbers (the GCD is 1). That's why I referred to it as an analogue. It is just a form of linear diophantine equation. From the section on linear diophantine equations (now the difference is called c): "It follows that there are also infinite solutions if c is a multiple of the greatest common divisor of a and b. If c is not a multiple of the greatest common divisor of a and b, then the Diophantine equation ax + by = c has no solutions." Of course, both this statement and some examples show that solutions exist. The problem is twofold: applying Bezout's identity and then multiplying through by two doesn't necessarily give the smallest solution, and the products ax and by are no longer odd.

The problem is finding integers with certain numbers of divisors. For example, 33 and 35 are the smallest consecutive odd numbers with 4 divisors each. I have an algorithm for generating tons of single numbers with this property, but generating all occurrences is a very inefficient way to find consecutive pairs. Bezout's identity provided a great tool when doing the problem for consecutive numbers (one even and one odd) since the difference of 1 satisfied the conditions of the identity. Walt (talk) 02:04, 13 July 2011 (UTC)

Probability problem; not your usual combinatorics problem

Hello, all. I've been working on this problem for some time, and am quite unsure of how to go about it properly, mainly due to my lack of knowledge in the topic. Anyway, enough with the irrelevant talk, here it is:

There are six boxes on a marry-go-round, each arranged so that they together form a circle on the merry-go-round that has been divided into six equal parts. Within the boxes are (let's use a colorful example) 1 boy, 2 girls, and 3 dogs. If the merry-go-round is spun once, and when it stops with a particular section (box) facing at a predetermined set, so that there are only six possible arrangements (think of the Wheel_of_Fortune_(U.S._game_show) and the mechanism used for that), then a box is opened and the item inside it is removed. Thus (here's the problem:), what are the chances that 1 boy, 1 girl, and 1 dog remain in their boxes after the merry-go-round has been spun 3 separate times?

The reason this isn't your normal combinatorics problem -- we can't just take the possible unordered, non-repeating set of three items -- is because there are only 3 spins permitted. In that case, one has to consider a whole new panoply of possibilities, which wouldn't be the case if the merry-go-round was spun just once with three boxes opened in some order.

Any ideas on how to tackle this properly? Numbility (talk) 20:19, 12 July 2011 (UTC)

My intuition tells me that your (presumed random) spins will treat each box alike, and so the problem can be considered to be the same as finding the chance that, given a random arrangement of one boy, two girls and three dogs, the first three will be, in some order, a girl and two dogs. Each of GDDxxx, DGDxxx and DDGxxx can arise in 6 ways (permuting BGD, which are the members of xxx), and so the total number of favourable ways is 18. Six objects of which two are alike and three are alike can be arranged in 60 ways, so your answer is 18/60 or 3/10.←86.155.185.195 (talk) 21:16, 12 July 2011 (UTC)

This was my initial guess, namely, computing the probability of the six possible arrangements (3dogs * 2girls * 1boy) divided by n!/r!(n-r)!, where r=3, the items chosen, and n=6, the total number of items. But as I said, the fact that there are three spins is a definite constraint that cannot be avoided. Thus, 6/20 = .3000. But I doubt that this is correct, which is why it's not your usual combinatorics problem. Numbility (talk) 23:16, 12 July 2011 (UTC)

On second thoughts, on each spin a particular box has 1 chance in 6 of being chosen. So the probability of the first three being, in order, GDD is (2/6)(3/6)(2/6) = 12/216. Likewise, the possibilities DGD and DDG also have probability 12/216. So the probability of some one of these events ocurring, and so leaving behind one each of boy, girl and dog, will be 3.12/216 = 1/6.←86.155.185.195 (talk) 21:33, 12 July 2011 (UTC)

Keep in mind that once a box has been opened it can still be a result in the following spins. It does not appear to me valid to successively multiply the probability of a box once it has been opened, because the kind of computation required will change. Thus, with each spin and an opening of a box, the problem will necessarily change and this has to be accounted for in the result. Numbility (talk) 23:16, 12 July 2011 (UTC)

The problem is small enough to just list the possible ways to be left with 1 of each. You can pick the same girl or dog 3 times in a row, which gives you 5 ways. You can pick a girl and a dog in 6 ways and each pair has 6 ways of being chosen (GGD, GDG, DGG, DDG, DGD, GDD). You can pick 3 pairs of dog and there are the same 6 ways for that pair to be chosen. There are 6 sets of 1 girl and 2 dogs, each of which can be chosen in 6 ways. So that should come out to 95/216 if I haven't done anything wrong. Maybe someone else has a more clever way to go about it though. Rckrone (talk) 01:50, 13 July 2011 (UTC)

I thought that, and used a pretty good method of finding out the bare probability by counting all of the possibilities by hand, as it were. However, since I don't know the formalisms involved, I'm not sure it's a valid result, which is why I'm trying to bounce this off you guys.

a=1 ; b=2 c=2 ; d=3 e=3 f=3
(Count the numbers next to the cases to obtain the result for each row.)
1st		{a,b,c,d,e}*3 [+3*2] {a,b,d,e,f}*2 [+2*2] {b,c,d,e,f}*1 [+1*2]				= 6 [+12] = 18
spin
result		One must take into account that the next spin may land on
		a previously emptied box. Thus add two self-returning paths here 
		for each unique case.
 
2nd		{a,b,c,d}*3 [+3] {a,b,d,e}*6 [+6] {a,d,e,f}*1 [+1] {b,c,d,e}*3 [+3] {b,d,e,f}*2 [+2]	= 15 [+15] = 30
spin
result		As above, but add one for one spin remaining.

3rd		{a,b,c}*1 >>>>{a,b,d}*6<<<< {a,d,e}*3 {d,e,f}*1 {b,c,d}*3 {b,d,e}*6			= 20
spin
result		"><" signifies the sought-for case (one of each item), of which there are six.

Therefore, 6/68 is the probability.

On that note, your calculation looks wrong to me. Anyone got better ideas? Numbility (talk) 03:49, 13 July 2011 (UTC)

Perhaps I am misunderstanding the problem or your answer, but I don't think you can pick the same girl three times in a row, because once a spin lands on her once, she is removed from her box and is not available to be chosen again. Also, if you happened to land on the same girl's box three times in a row with your three spins, you would be left with one boy, one girl, and three dogs in the boxes on the merry-go-round, which is not the desired outcome. —Bkell (talk) 03:06, 13 July 2011 (UTC)

It should be pretty clear that there are only six ways in which there can be 1 boy, 1 girl, and 1 dog remaining. The issue is figuring out all of the general cases in which these 6 occur. Numbility (talk) 03:57, 13 July 2011 (UTC)

I interpreted the question to mean you want at least 1 boy, 1 girl and, 1 dog left. If you want exactly those left you just consider the last case of picking 2 dogs and a girl which as I mentioned has 36 ways of happening, so 1/6. Rckrone (talk) 03:39, 13 July 2011 (UTC)

No, what are the chances that 1 boy, 1 dog, and 1 girl will remain after only three spins? Look at my previous answer above to see the "bare bones" of the problem. Numbility (talk) 03:53, 13 July 2011 (UTC)

If I'm understanding the problem correctly, your three spins must land on dogs twice and a girl once, not necessarily in that order. So, the three spins could land girl–dog–dog, dog–girl–dog, or dog–dog–girl. We just need to compute the probabilities of each of those three and add them together (since they are mutually exclusive). What is the probability of getting girl–dog–dog on the three spins? The probability of landing on a girl on the first spin is 2/6, or 1/3; after that, the probability of landing on a dog on the next spin is 3/6, or 1/2; and after that, since there are only two dogs remaining, the probability of landing on a dog on the last spin is 2/6, or 1/3. So the probability of getting girl–dog–dog is (1/3)×(1/2)×(1/3)=1/12. Similarly, in order to get dog–girl–dog, the first spin must land on a dog, which will happen with probability 3/6=1/2; the second must land on a girl, with probability 2/6=1/3; and the last spin must land on one of the two remaining dogs, with probability 2/6=1/3. So the probability of getting dog–girl–dog is again (1/2)×(1/3)×(1/3)=1/12. If you work out the probability of getting dog–dog–girl like this, you'll see that it is also 1/12. So the overall probability of a desirable outcome is (1/12)+(1/12)+(1/12)=3/12=1/4. —Bkell (talk) 02:59, 13 July 2011 (UTC)

No, it's not that they "must" land that way. They can land any which way they please, even on an empty box, so the calculation has to take that into account. Numbility (talk) 03:53, 13 July 2011 (UTC)

Bkell means that they must land that way in order to reach the outcome you want: Exactly 1 boy, 1 girl, and 1 dog remain. However, Bkell incorrectly wrote (1/3)×(1/2)×(1/3)=1/12. But (1/3)×(1/2)×(1/3)=1/18. Then the total probability of success becomes 3/18 = 1/6, as 86.155.185.195 and Rckrone have already said. PrimeHunter (talk) 04:28, 13 July 2011 (UTC)

Oh, yep, I can't multiply. Thanks. —Bkell (talk) 12:06, 13 July 2011 (UTC)

@PrimeHunter (or anyone), so how do I relate my crude analysis, which gave me 6/68 or 2/34, to this 1/6 probability? Where am I wrong? Numbility (talk) 15:16, 13 July 2011 (UTC)

I think I may be beginning to see the light, and I hope you guys will help me on this. Looking at my table of scenarios, I've come to the realization that I should discard the undesired outcomes. Thus, at the first spin, I have only 5 of 6 total cases that I want, at the second, I have only 9 of 15 that I want, and at the third only 6 of 20. At this stage, I take the fractions of each ordinal spin-result and multiply them: (5/6)(9/15)(6/20) = 60/1800 which reduces to 3/20. This answer is surprisingly close to 1/6 and seems to make a lot more sense than 1/6. What say you all? Numbility (talk) 15:55, 13 July 2011 (UTC)

I don't understand your analysis. Can you explain it a little more thoroughly, in words? —Bkell (talk) 19:58, 13 July 2011 (UTC)

Okay, after I've reread what you've written, let me say a little bit about what I think you're doing. You are correct when you say that your first spin will lead toward a desirable outcome 5/6 of the time (the other 1/6 of the time the first spin will land on a boy, and then certainly the desired outcome will not occur, no matter what happens in the next two spins). However, the probability that your second spin will continue toward a desirable outcome depends on what exactly happened in the first spin. If the first spin landed on a girl, then the second spin must land on a dog to lead toward a desirable outcome, and the probability of that happening is 3/6. On the other hand, if the first spin landed on a dog, then the second spin could land either on another dog or on a girl and still lead toward a desirable outcome, so the probability is 4/6. So you can't just multiply the whole 5/6 from the first spin by the probability of "success" in the second spin (where "success" means "leading toward a desirable outcome"), because that second probability is not independent of what happened in the first spin. Instead, you should split up that 5/6 from the first spin like this: With probability 3/6, the first spin will land on a dog, and then, if that happens, the probability of success in the second spin is 4/6; the probability of both of these things happening in the first two spins is (3/6)×(4/6)=1/3. With probability 2/6, the first spin will land on a girl, and then, if that happens, the probability of success in the second spin is 3/6; the probability of both of these things happening in the first two spins is (2/6)×(3/6)=1/6. Then add the probabilities of these two mutually exclusive events to get the total probability of success through the first two spins: (1/3)+(1/6)=1/2. Now, to finish this analysis, you must split up the 4/6 probability of success in the second spin following a dog in the first spin, for the same reason: After the first spin lands on a dog, and a "successful" second spin, the probability of success in the third spin depends on what exactly happened in the second spin (dog or girl). This analysis is what I was doing above. —Bkell (talk) 20:13, 13 July 2011 (UTC)

I guess, in this particular problem, after the first two spins are "dog" and "success" (i.e., either "dog–dog" or "dog–girl"), the probability of success in the third spin happens to be the same (2/6) whether the second spin was a dog or a girl. In general, though, when dealing with conditional probabilities like this (where the probability of an event B is different depending on whether or not some other event A happened), you need to consider different cases for the event A: one line of reasoning for the case when A occurs, and another for the case when A does not occur. This is called the law of total probability. —Bkell (talk) 20:41, 13 July 2011 (UTC)

Maybe I'm dense (others seem to have quickly agreed that the answer is 1/6, but I'm not sure), but I'm having a hard time seeing how the logic of conditional probability applies to this problem. Perhaps a simplified version of the problem would help. Going back to the basics, let's say I have four cards: 2 spades, 1 heart, 1 diamond. After the deck has been shuffled, a card is withdrawn and a blank is inserted in its place, and the deck is shuffled again. What is the probability of there being 1 spade and 1 heart after 2 shuffles and withdrawals? So we have:

a=1, b=2, c=3, d=3
{a=heart, b=diamond, c=d=spade}
1st	(one blank	{a,b,c}*2	{b,c,d}*1	{a,c,d}*1			=	4 cases
	for all cases)	(good)		(bad)		(good)

2nd	(two blanks	{b,c}*2		{a,b}*1		{c,d}*1		>>{a,c}*2<<		=	6 cases
	for all cases)	(bad)		(bad)		(bad)		(good)

1st has 3 good cases out of 4, thus, 3/4; 2nd has 2 good cases out of 6, thus, 1/3. The total probability
is (3/4)(1/3)= 3/12 = 1/4.

What is the flaw with this analysis as we consider the "good" cases in the numerator over the total cases in the denominator? It should be clear that in the good cases, the result we're after is a possibility therein, and so we can multiply the good cases and divide this by the product of individual ordinal cases. This problem is fundamentally the same as my earlier problem with a few simplifications (fewer cases, so hopefully less confusion). Thanks for your time. Numbility (talk) 03:17, 14 July 2011 (UTC)

In addition, if you look closely, we don't interpret the 2nd row as being "conditional on" the first result, nor do we consider the 1st row as being "conditional on" the random arrangement of the cards at the initial condition. It seems pretty clear to me that in the merry-go-round version, even a 3rd row of results aren't "conditional on" what happened in the 2nd row, which means I can divide the product of good cases by the product of all cases that exist. So 3/20 must be the answer, not 1/6. Numbility (talk) 03:37, 14 July 2011 (UTC)

The answer can't be 3/20. Here's why. There are 6 possible outcomes for the first spin (one for each box). Then, for each of those possible outcomes (and independently of the first outcome), there are 6 possible outcomes for the second spin. Likewise there are 6 possible outcomes for the third spin. This gives 6×6×6=216 possible outcomes for the three spins together. All of these 216 outcomes are equally likely. So the probability that you get a desired outcome has got to be x/216 for some value of x. No such fraction can simplify to 3/20, because 20 has a factor of 5 and 216 doesn't. —Bkell (talk) 04:01, 14 July 2011 (UTC)

I count 15 unique cases at the second spin and 20 at the 3rd. So there are a total of 1,800 unique cases within the "space" of the merry-go-round, not 216. Numbility (talk) 04:06, 14 July 2011 (UTC)

Here are the 216 possible ways the three spins could turn out. I've stereotypically named the boy Johnny, the girls Mary and Susan, and the dogs Fido, Rover, and Spot. If a spin lands on someone's box who was already chosen in a previous spin, that box is empty.

216 possible outcomes of the three spins

1. Spins land on Johnny, Johnny (now empty), Johnny (now empty). Those remaining in boxes after 3 spins: Mary, Susan, Fido, Rover, Spot. BAD.
2. Spins land on Johnny, Johnny (now empty), Mary. Those remaining in boxes after 3 spins: Susan, Fido, Rover, Spot. BAD.
3. Spins land on Johnny, Johnny (now empty), Susan. Those remaining in boxes after 3 spins: Mary, Fido, Rover, Spot. BAD.
4. Spins land on Johnny, Johnny (now empty), Fido. Those remaining in boxes after 3 spins: Mary, Susan, Rover, Spot. BAD.
5. Spins land on Johnny, Johnny (now empty), Rover. Those remaining in boxes after 3 spins: Mary, Susan, Fido, Spot. BAD.
6. Spins land on Johnny, Johnny (now empty), Spot. Those remaining in boxes after 3 spins: Mary, Susan, Fido, Rover. BAD.
7. Spins land on Johnny, Mary, Johnny (now empty). Those remaining in boxes after 3 spins: Susan, Fido, Rover, Spot. BAD.
8. Spins land on Johnny, Mary, Mary (now empty). Those remaining in boxes after 3 spins: Susan, Fido, Rover, Spot. BAD.
9. Spins land on Johnny, Mary, Susan. Those remaining in boxes after 3 spins: Fido, Rover, Spot. BAD.
10. Spins land on Johnny, Mary, Fido. Those remaining in boxes after 3 spins: Susan, Rover, Spot. BAD.
11. Spins land on Johnny, Mary, Rover. Those remaining in boxes after 3 spins: Susan, Fido, Spot. BAD.
12. Spins land on Johnny, Mary, Spot. Those remaining in boxes after 3 spins: Susan, Fido, Rover. BAD.
13. Spins land on Johnny, Susan, Johnny (now empty). Those remaining in boxes after 3 spins: Mary, Fido, Rover, Spot. BAD.
14. Spins land on Johnny, Susan, Mary. Those remaining in boxes after 3 spins: Fido, Rover, Spot. BAD.
15. Spins land on Johnny, Susan, Susan (now empty). Those remaining in boxes after 3 spins: Mary, Fido, Rover, Spot. BAD.
16. Spins land on Johnny, Susan, Fido. Those remaining in boxes after 3 spins: Mary, Rover, Spot. BAD.
17. Spins land on Johnny, Susan, Rover. Those remaining in boxes after 3 spins: Mary, Fido, Spot. BAD.
18. Spins land on Johnny, Susan, Spot. Those remaining in boxes after 3 spins: Mary, Fido, Rover. BAD.
19. Spins land on Johnny, Fido, Johnny (now empty). Those remaining in boxes after 3 spins: Mary, Susan, Rover, Spot. BAD.
20. Spins land on Johnny, Fido, Mary. Those remaining in boxes after 3 spins: Susan, Rover, Spot. BAD.
21. Spins land on Johnny, Fido, Susan. Those remaining in boxes after 3 spins: Mary, Rover, Spot. BAD.
22. Spins land on Johnny, Fido, Fido (now empty). Those remaining in boxes after 3 spins: Mary, Susan, Rover, Spot. BAD.
23. Spins land on Johnny, Fido, Rover. Those remaining in boxes after 3 spins: Mary, Susan, Spot. BAD.
24. Spins land on Johnny, Fido, Spot. Those remaining in boxes after 3 spins: Mary, Susan, Rover. BAD.
25. Spins land on Johnny, Rover, Johnny (now empty). Those remaining in boxes after 3 spins: Mary, Susan, Fido, Spot. BAD.
26. Spins land on Johnny, Rover, Mary. Those remaining in boxes after 3 spins: Susan, Fido, Spot. BAD.
27. Spins land on Johnny, Rover, Susan. Those remaining in boxes after 3 spins: Mary, Fido, Spot. BAD.
28. Spins land on Johnny, Rover, Fido. Those remaining in boxes after 3 spins: Mary, Susan, Spot. BAD.
29. Spins land on Johnny, Rover, Rover (now empty). Those remaining in boxes after 3 spins: Mary, Susan, Fido, Spot. BAD.
30. Spins land on Johnny, Rover, Spot. Those remaining in boxes after 3 spins: Mary, Susan, Fido. BAD.
31. Spins land on Johnny, Spot, Johnny (now empty). Those remaining in boxes after 3 spins: Mary, Susan, Fido, Rover. BAD.
32. Spins land on Johnny, Spot, Mary. Those remaining in boxes after 3 spins: Susan, Fido, Rover. BAD.
33. Spins land on Johnny, Spot, Susan. Those remaining in boxes after 3 spins: Mary, Fido, Rover. BAD.
34. Spins land on Johnny, Spot, Fido. Those remaining in boxes after 3 spins: Mary, Susan, Rover. BAD.
35. Spins land on Johnny, Spot, Rover. Those remaining in boxes after 3 spins: Mary, Susan, Fido. BAD.
36. Spins land on Johnny, Spot, Spot (now empty). Those remaining in boxes after 3 spins: Mary, Susan, Fido, Rover. BAD.
37. Spins land on Mary, Johnny, Johnny (now empty). Those remaining in boxes after 3 spins: Susan, Fido, Rover, Spot. BAD.
38. Spins land on Mary, Johnny, Mary (now empty). Those remaining in boxes after 3 spins: Susan, Fido, Rover, Spot. BAD.
39. Spins land on Mary, Johnny, Susan. Those remaining in boxes after 3 spins: Fido, Rover, Spot. BAD.
40. Spins land on Mary, Johnny, Fido. Those remaining in boxes after 3 spins: Susan, Rover, Spot. BAD.
41. Spins land on Mary, Johnny, Rover. Those remaining in boxes after 3 spins: Susan, Fido, Spot. BAD.
42. Spins land on Mary, Johnny, Spot. Those remaining in boxes after 3 spins: Susan, Fido, Rover. BAD.
43. Spins land on Mary, Mary (now empty), Johnny. Those remaining in boxes after 3 spins: Susan, Fido, Rover, Spot. BAD.
44. Spins land on Mary, Mary (now empty), Mary (now empty). Those remaining in boxes after 3 spins: Johnny, Susan, Fido, Rover, Spot. BAD.
45. Spins land on Mary, Mary (now empty), Susan. Those remaining in boxes after 3 spins: Johnny, Fido, Rover, Spot. BAD.
46. Spins land on Mary, Mary (now empty), Fido. Those remaining in boxes after 3 spins: Johnny, Susan, Rover, Spot. BAD.
47. Spins land on Mary, Mary (now empty), Rover. Those remaining in boxes after 3 spins: Johnny, Susan, Fido, Spot. BAD.
48. Spins land on Mary, Mary (now empty), Spot. Those remaining in boxes after 3 spins: Johnny, Susan, Fido, Rover. BAD.
49. Spins land on Mary, Susan, Johnny. Those remaining in boxes after 3 spins: Fido, Rover, Spot. BAD.
50. Spins land on Mary, Susan, Mary (now empty). Those remaining in boxes after 3 spins: Johnny, Fido, Rover, Spot. BAD.
51. Spins land on Mary, Susan, Susan (now empty). Those remaining in boxes after 3 spins: Johnny, Fido, Rover, Spot. BAD.
52. Spins land on Mary, Susan, Fido. Those remaining in boxes after 3 spins: Johnny, Rover, Spot. BAD.
53. Spins land on Mary, Susan, Rover. Those remaining in boxes after 3 spins: Johnny, Fido, Spot. BAD.
54. Spins land on Mary, Susan, Spot. Those remaining in boxes after 3 spins: Johnny, Fido, Rover. BAD.
55. Spins land on Mary, Fido, Johnny. Those remaining in boxes after 3 spins: Susan, Rover, Spot. BAD.
56. Spins land on Mary, Fido, Mary (now empty). Those remaining in boxes after 3 spins: Johnny, Susan, Rover, Spot. BAD.
57. Spins land on Mary, Fido, Susan. Those remaining in boxes after 3 spins: Johnny, Rover, Spot. BAD.
58. Spins land on Mary, Fido, Fido (now empty). Those remaining in boxes after 3 spins: Johnny, Susan, Rover, Spot. BAD.
59. Spins land on Mary, Fido, Rover. Those remaining in boxes after 3 spins: Johnny, Susan, Spot. GOOD.
60. Spins land on Mary, Fido, Spot. Those remaining in boxes after 3 spins: Johnny, Susan, Rover. GOOD.
61. Spins land on Mary, Rover, Johnny. Those remaining in boxes after 3 spins: Susan, Fido, Spot. BAD.
62. Spins land on Mary, Rover, Mary (now empty). Those remaining in boxes after 3 spins: Johnny, Susan, Fido, Spot. BAD.
63. Spins land on Mary, Rover, Susan. Those remaining in boxes after 3 spins: Johnny, Fido, Spot. BAD.
64. Spins land on Mary, Rover, Fido. Those remaining in boxes after 3 spins: Johnny, Susan, Spot. GOOD.
65. Spins land on Mary, Rover, Rover (now empty). Those remaining in boxes after 3 spins: Johnny, Susan, Fido, Spot. BAD.
66. Spins land on Mary, Rover, Spot. Those remaining in boxes after 3 spins: Johnny, Susan, Fido. GOOD.
67. Spins land on Mary, Spot, Johnny. Those remaining in boxes after 3 spins: Susan, Fido, Rover. BAD.
68. Spins land on Mary, Spot, Mary (now empty). Those remaining in boxes after 3 spins: Johnny, Susan, Fido, Rover. BAD.
69. Spins land on Mary, Spot, Susan. Those remaining in boxes after 3 spins: Johnny, Fido, Rover. BAD.
70. Spins land on Mary, Spot, Fido. Those remaining in boxes after 3 spins: Johnny, Susan, Rover. GOOD.
71. Spins land on Mary, Spot, Rover. Those remaining in boxes after 3 spins: Johnny, Susan, Fido. GOOD.
72. Spins land on Mary, Spot, Spot (now empty). Those remaining in boxes after 3 spins: Johnny, Susan, Fido, Rover. BAD.
73. Spins land on Susan, Johnny, Johnny (now empty). Those remaining in boxes after 3 spins: Mary, Fido, Rover, Spot. BAD.
74. Spins land on Susan, Johnny, Mary. Those remaining in boxes after 3 spins: Fido, Rover, Spot. BAD.
75. Spins land on Susan, Johnny, Susan (now empty). Those remaining in boxes after 3 spins: Mary, Fido, Rover, Spot. BAD.
76. Spins land on Susan, Johnny, Fido. Those remaining in boxes after 3 spins: Mary, Rover, Spot. BAD.
77. Spins land on Susan, Johnny, Rover. Those remaining in boxes after 3 spins: Mary, Fido, Spot. BAD.
78. Spins land on Susan, Johnny, Spot. Those remaining in boxes after 3 spins: Mary, Fido, Rover. BAD.
79. Spins land on Susan, Mary, Johnny. Those remaining in boxes after 3 spins: Fido, Rover, Spot. BAD.
80. Spins land on Susan, Mary, Mary (now empty). Those remaining in boxes after 3 spins: Johnny, Fido, Rover, Spot. BAD.
81. Spins land on Susan, Mary, Susan (now empty). Those remaining in boxes after 3 spins: Johnny, Fido, Rover, Spot. BAD.
82. Spins land on Susan, Mary, Fido. Those remaining in boxes after 3 spins: Johnny, Rover, Spot. BAD.
83. Spins land on Susan, Mary, Rover. Those remaining in boxes after 3 spins: Johnny, Fido, Spot. BAD.
84. Spins land on Susan, Mary, Spot. Those remaining in boxes after 3 spins: Johnny, Fido, Rover. BAD.
85. Spins land on Susan, Susan (now empty), Johnny. Those remaining in boxes after 3 spins: Mary, Fido, Rover, Spot. BAD.
86. Spins land on Susan, Susan (now empty), Mary. Those remaining in boxes after 3 spins: Johnny, Fido, Rover, Spot. BAD.
87. Spins land on Susan, Susan (now empty), Susan (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Fido, Rover, Spot. BAD.
88. Spins land on Susan, Susan (now empty), Fido. Those remaining in boxes after 3 spins: Johnny, Mary, Rover, Spot. BAD.
89. Spins land on Susan, Susan (now empty), Rover. Those remaining in boxes after 3 spins: Johnny, Mary, Fido, Spot. BAD.
90. Spins land on Susan, Susan (now empty), Spot. Those remaining in boxes after 3 spins: Johnny, Mary, Fido, Rover. BAD.
91. Spins land on Susan, Fido, Johnny. Those remaining in boxes after 3 spins: Mary, Rover, Spot. BAD.
92. Spins land on Susan, Fido, Mary. Those remaining in boxes after 3 spins: Johnny, Rover, Spot. BAD.
93. Spins land on Susan, Fido, Susan (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Rover, Spot. BAD.
94. Spins land on Susan, Fido, Fido (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Rover, Spot. BAD.
95. Spins land on Susan, Fido, Rover. Those remaining in boxes after 3 spins: Johnny, Mary, Spot. GOOD.
96. Spins land on Susan, Fido, Spot. Those remaining in boxes after 3 spins: Johnny, Mary, Rover. GOOD.
97. Spins land on Susan, Rover, Johnny. Those remaining in boxes after 3 spins: Mary, Fido, Spot. BAD.
98. Spins land on Susan, Rover, Mary. Those remaining in boxes after 3 spins: Johnny, Fido, Spot. BAD.
99. Spins land on Susan, Rover, Susan (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Fido, Spot. BAD.
100. Spins land on Susan, Rover, Fido. Those remaining in boxes after 3 spins: Johnny, Mary, Spot. GOOD.
101. Spins land on Susan, Rover, Rover (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Fido, Spot. BAD.
102. Spins land on Susan, Rover, Spot. Those remaining in boxes after 3 spins: Johnny, Mary, Fido. GOOD.
103. Spins land on Susan, Spot, Johnny. Those remaining in boxes after 3 spins: Mary, Fido, Rover. BAD.
104. Spins land on Susan, Spot, Mary. Those remaining in boxes after 3 spins: Johnny, Fido, Rover. BAD.
105. Spins land on Susan, Spot, Susan (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Fido, Rover. BAD.
106. Spins land on Susan, Spot, Fido. Those remaining in boxes after 3 spins: Johnny, Mary, Rover. GOOD.
107. Spins land on Susan, Spot, Rover. Those remaining in boxes after 3 spins: Johnny, Mary, Fido. GOOD.
108. Spins land on Susan, Spot, Spot (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Fido, Rover. BAD.
109. Spins land on Fido, Johnny, Johnny (now empty). Those remaining in boxes after 3 spins: Mary, Susan, Rover, Spot. BAD.
110. Spins land on Fido, Johnny, Mary. Those remaining in boxes after 3 spins: Susan, Rover, Spot. BAD.
111. Spins land on Fido, Johnny, Susan. Those remaining in boxes after 3 spins: Mary, Rover, Spot. BAD.
112. Spins land on Fido, Johnny, Fido (now empty). Those remaining in boxes after 3 spins: Mary, Susan, Rover, Spot. BAD.
113. Spins land on Fido, Johnny, Rover. Those remaining in boxes after 3 spins: Mary, Susan, Spot. BAD.
114. Spins land on Fido, Johnny, Spot. Those remaining in boxes after 3 spins: Mary, Susan, Rover. BAD.
115. Spins land on Fido, Mary, Johnny. Those remaining in boxes after 3 spins: Susan, Rover, Spot. BAD.
116. Spins land on Fido, Mary, Mary (now empty). Those remaining in boxes after 3 spins: Johnny, Susan, Rover, Spot. BAD.
117. Spins land on Fido, Mary, Susan. Those remaining in boxes after 3 spins: Johnny, Rover, Spot. BAD.
118. Spins land on Fido, Mary, Fido (now empty). Those remaining in boxes after 3 spins: Johnny, Susan, Rover, Spot. BAD.
119. Spins land on Fido, Mary, Rover. Those remaining in boxes after 3 spins: Johnny, Susan, Spot. GOOD.
120. Spins land on Fido, Mary, Spot. Those remaining in boxes after 3 spins: Johnny, Susan, Rover. GOOD.
121. Spins land on Fido, Susan, Johnny. Those remaining in boxes after 3 spins: Mary, Rover, Spot. BAD.
122. Spins land on Fido, Susan, Mary. Those remaining in boxes after 3 spins: Johnny, Rover, Spot. BAD.
123. Spins land on Fido, Susan, Susan (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Rover, Spot. BAD.
124. Spins land on Fido, Susan, Fido (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Rover, Spot. BAD.
125. Spins land on Fido, Susan, Rover. Those remaining in boxes after 3 spins: Johnny, Mary, Spot. GOOD.
126. Spins land on Fido, Susan, Spot. Those remaining in boxes after 3 spins: Johnny, Mary, Rover. GOOD.
127. Spins land on Fido, Fido (now empty), Johnny. Those remaining in boxes after 3 spins: Mary, Susan, Rover, Spot. BAD.
128. Spins land on Fido, Fido (now empty), Mary. Those remaining in boxes after 3 spins: Johnny, Susan, Rover, Spot. BAD.
129. Spins land on Fido, Fido (now empty), Susan. Those remaining in boxes after 3 spins: Johnny, Mary, Rover, Spot. BAD.
130. Spins land on Fido, Fido (now empty), Fido (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Rover, Spot. BAD.
131. Spins land on Fido, Fido (now empty), Rover. Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Spot. BAD.
132. Spins land on Fido, Fido (now empty), Spot. Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Rover. BAD.
133. Spins land on Fido, Rover, Johnny. Those remaining in boxes after 3 spins: Mary, Susan, Spot. BAD.
134. Spins land on Fido, Rover, Mary. Those remaining in boxes after 3 spins: Johnny, Susan, Spot. GOOD.
135. Spins land on Fido, Rover, Susan. Those remaining in boxes after 3 spins: Johnny, Mary, Spot. GOOD.
136. Spins land on Fido, Rover, Fido (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Spot. BAD.
137. Spins land on Fido, Rover, Rover (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Spot. BAD.
138. Spins land on Fido, Rover, Spot. Those remaining in boxes after 3 spins: Johnny, Mary, Susan. BAD.
139. Spins land on Fido, Spot, Johnny. Those remaining in boxes after 3 spins: Mary, Susan, Rover. BAD.
140. Spins land on Fido, Spot, Mary. Those remaining in boxes after 3 spins: Johnny, Susan, Rover. GOOD.
141. Spins land on Fido, Spot, Susan. Those remaining in boxes after 3 spins: Johnny, Mary, Rover. GOOD.
142. Spins land on Fido, Spot, Fido (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Rover. BAD.
143. Spins land on Fido, Spot, Rover. Those remaining in boxes after 3 spins: Johnny, Mary, Susan. BAD.
144. Spins land on Fido, Spot, Spot (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Rover. BAD.
145. Spins land on Rover, Johnny, Johnny (now empty). Those remaining in boxes after 3 spins: Mary, Susan, Fido, Spot. BAD.
146. Spins land on Rover, Johnny, Mary. Those remaining in boxes after 3 spins: Susan, Fido, Spot. BAD.
147. Spins land on Rover, Johnny, Susan. Those remaining in boxes after 3 spins: Mary, Fido, Spot. BAD.
148. Spins land on Rover, Johnny, Fido. Those remaining in boxes after 3 spins: Mary, Susan, Spot. BAD.
149. Spins land on Rover, Johnny, Rover (now empty). Those remaining in boxes after 3 spins: Mary, Susan, Fido, Spot. BAD.
150. Spins land on Rover, Johnny, Spot. Those remaining in boxes after 3 spins: Mary, Susan, Fido. BAD.
151. Spins land on Rover, Mary, Johnny. Those remaining in boxes after 3 spins: Susan, Fido, Spot. BAD.
152. Spins land on Rover, Mary, Mary (now empty). Those remaining in boxes after 3 spins: Johnny, Susan, Fido, Spot. BAD.
153. Spins land on Rover, Mary, Susan. Those remaining in boxes after 3 spins: Johnny, Fido, Spot. BAD.
154. Spins land on Rover, Mary, Fido. Those remaining in boxes after 3 spins: Johnny, Susan, Spot. GOOD.
155. Spins land on Rover, Mary, Rover (now empty). Those remaining in boxes after 3 spins: Johnny, Susan, Fido, Spot. BAD.
156. Spins land on Rover, Mary, Spot. Those remaining in boxes after 3 spins: Johnny, Susan, Fido. GOOD.
157. Spins land on Rover, Susan, Johnny. Those remaining in boxes after 3 spins: Mary, Fido, Spot. BAD.
158. Spins land on Rover, Susan, Mary. Those remaining in boxes after 3 spins: Johnny, Fido, Spot. BAD.
159. Spins land on Rover, Susan, Susan (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Fido, Spot. BAD.
160. Spins land on Rover, Susan, Fido. Those remaining in boxes after 3 spins: Johnny, Mary, Spot. GOOD.
161. Spins land on Rover, Susan, Rover (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Fido, Spot. BAD.
162. Spins land on Rover, Susan, Spot. Those remaining in boxes after 3 spins: Johnny, Mary, Fido. GOOD.
163. Spins land on Rover, Fido, Johnny. Those remaining in boxes after 3 spins: Mary, Susan, Spot. BAD.
164. Spins land on Rover, Fido, Mary. Those remaining in boxes after 3 spins: Johnny, Susan, Spot. GOOD.
165. Spins land on Rover, Fido, Susan. Those remaining in boxes after 3 spins: Johnny, Mary, Spot. GOOD.
166. Spins land on Rover, Fido, Fido (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Spot. BAD.
167. Spins land on Rover, Fido, Rover (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Spot. BAD.
168. Spins land on Rover, Fido, Spot. Those remaining in boxes after 3 spins: Johnny, Mary, Susan. BAD.
169. Spins land on Rover, Rover (now empty), Johnny. Those remaining in boxes after 3 spins: Mary, Susan, Fido, Spot. BAD.
170. Spins land on Rover, Rover (now empty), Mary. Those remaining in boxes after 3 spins: Johnny, Susan, Fido, Spot. BAD.
171. Spins land on Rover, Rover (now empty), Susan. Those remaining in boxes after 3 spins: Johnny, Mary, Fido, Spot. BAD.
172. Spins land on Rover, Rover (now empty), Fido. Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Spot. BAD.
173. Spins land on Rover, Rover (now empty), Rover (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Fido, Spot. BAD.
174. Spins land on Rover, Rover (now empty), Spot. Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Fido. BAD.
175. Spins land on Rover, Spot, Johnny. Those remaining in boxes after 3 spins: Mary, Susan, Fido. BAD.
176. Spins land on Rover, Spot, Mary. Those remaining in boxes after 3 spins: Johnny, Susan, Fido. GOOD.
177. Spins land on Rover, Spot, Susan. Those remaining in boxes after 3 spins: Johnny, Mary, Fido. GOOD.
178. Spins land on Rover, Spot, Fido. Those remaining in boxes after 3 spins: Johnny, Mary, Susan. BAD.
179. Spins land on Rover, Spot, Rover (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Fido. BAD.
180. Spins land on Rover, Spot, Spot (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Fido. BAD.
181. Spins land on Spot, Johnny, Johnny (now empty). Those remaining in boxes after 3 spins: Mary, Susan, Fido, Rover. BAD.
182. Spins land on Spot, Johnny, Mary. Those remaining in boxes after 3 spins: Susan, Fido, Rover. BAD.
183. Spins land on Spot, Johnny, Susan. Those remaining in boxes after 3 spins: Mary, Fido, Rover. BAD.
184. Spins land on Spot, Johnny, Fido. Those remaining in boxes after 3 spins: Mary, Susan, Rover. BAD.
185. Spins land on Spot, Johnny, Rover. Those remaining in boxes after 3 spins: Mary, Susan, Fido. BAD.
186. Spins land on Spot, Johnny, Spot (now empty). Those remaining in boxes after 3 spins: Mary, Susan, Fido, Rover. BAD.
187. Spins land on Spot, Mary, Johnny. Those remaining in boxes after 3 spins: Susan, Fido, Rover. BAD.
188. Spins land on Spot, Mary, Mary (now empty). Those remaining in boxes after 3 spins: Johnny, Susan, Fido, Rover. BAD.
189. Spins land on Spot, Mary, Susan. Those remaining in boxes after 3 spins: Johnny, Fido, Rover. BAD.
190. Spins land on Spot, Mary, Fido. Those remaining in boxes after 3 spins: Johnny, Susan, Rover. GOOD.
191. Spins land on Spot, Mary, Rover. Those remaining in boxes after 3 spins: Johnny, Susan, Fido. GOOD.
192. Spins land on Spot, Mary, Spot (now empty). Those remaining in boxes after 3 spins: Johnny, Susan, Fido, Rover. BAD.
193. Spins land on Spot, Susan, Johnny. Those remaining in boxes after 3 spins: Mary, Fido, Rover. BAD.
194. Spins land on Spot, Susan, Mary. Those remaining in boxes after 3 spins: Johnny, Fido, Rover. BAD.
195. Spins land on Spot, Susan, Susan (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Fido, Rover. BAD.
196. Spins land on Spot, Susan, Fido. Those remaining in boxes after 3 spins: Johnny, Mary, Rover. GOOD.
197. Spins land on Spot, Susan, Rover. Those remaining in boxes after 3 spins: Johnny, Mary, Fido. GOOD.
198. Spins land on Spot, Susan, Spot (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Fido, Rover. BAD.
199. Spins land on Spot, Fido, Johnny. Those remaining in boxes after 3 spins: Mary, Susan, Rover. BAD.
200. Spins land on Spot, Fido, Mary. Those remaining in boxes after 3 spins: Johnny, Susan, Rover. GOOD.
201. Spins land on Spot, Fido, Susan. Those remaining in boxes after 3 spins: Johnny, Mary, Rover. GOOD.
202. Spins land on Spot, Fido, Fido (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Rover. BAD.
203. Spins land on Spot, Fido, Rover. Those remaining in boxes after 3 spins: Johnny, Mary, Susan. BAD.
204. Spins land on Spot, Fido, Spot (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Rover. BAD.
205. Spins land on Spot, Rover, Johnny. Those remaining in boxes after 3 spins: Mary, Susan, Fido. BAD.
206. Spins land on Spot, Rover, Mary. Those remaining in boxes after 3 spins: Johnny, Susan, Fido. GOOD.
207. Spins land on Spot, Rover, Susan. Those remaining in boxes after 3 spins: Johnny, Mary, Fido. GOOD.
208. Spins land on Spot, Rover, Fido. Those remaining in boxes after 3 spins: Johnny, Mary, Susan. BAD.
209. Spins land on Spot, Rover, Rover (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Fido. BAD.
210. Spins land on Spot, Rover, Spot (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Fido. BAD.
211. Spins land on Spot, Spot (now empty), Johnny. Those remaining in boxes after 3 spins: Mary, Susan, Fido, Rover. BAD.
212. Spins land on Spot, Spot (now empty), Mary. Those remaining in boxes after 3 spins: Johnny, Susan, Fido, Rover. BAD.
213. Spins land on Spot, Spot (now empty), Susan. Those remaining in boxes after 3 spins: Johnny, Mary, Fido, Rover. BAD.
214. Spins land on Spot, Spot (now empty), Fido. Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Rover. BAD.
215. Spins land on Spot, Spot (now empty), Rover. Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Fido. BAD.
216. Spins land on Spot, Spot (now empty), Spot (now empty). Those remaining in boxes after 3 spins: Johnny, Mary, Susan, Fido, Rover. BAD.

I don't have time just this minute to improve the program I wrote to list these so that it labels them as good or bad outcomes, but if you count them you'll see that 36 of these result in a good outcome, so the probability is 36/216, or 1/6. —Bkell (talk) 04:43, 14 July 2011 (UTC)

Okay, now that I have a few minutes, I've improved my little program so that it more clearly states, for each of the 216 equally possible outcomes, where the spins land, who is remaining in their boxes after the three spins, and whether this outcome is good (exactly one boy, one girl, and one dog remain) or bad (otherwise). If you count up the GOOD rows, you will see that there are 36 of them. —Bkell (talk) 06:25, 14 July 2011 (UTC)

Your answer of 1/4 happens to be correct for this problem, I believe, but I don't see why your analysis makes sense. For the first draw, you have {a,b,c}*2; I assume this is shorthand for {a,b,c} and {a,b,d} because those two cases are equivalent. Likewise, for the second draw, you have {b,c}*2, which I assume means {b,c} and {b,d}, and you have {a,c}*2, which I take to mean {a,c} and {a,d}. You say, for the second draw, "6 cases for all cases"—but that's not true; you don't have the same six possibilities for all cases. If the first draw resulted in {a,b,c}, for example, you can't get {b,d} after the second draw, because you've removed d from the deck. So {b,c} should count only once in that case. The same goes for {a,c}, and {c,d} can't happen at all—and you forgot the possibility that after the second draw you might be left with {a,b,c} because you drew the blank. So, if the first draw leaves you with {a,b,c}, then there are only four possibilities, not six, for what you are left with after the second draw: {a,b}, {a,c}, {b,c}, and {a,b,c}. One of these four, {a,c}, is a good outcome. The probability of getting to {a,c} in this manner equals the probability of getting {a,b,c} after the first draw (which is 1/4) multiplied by the probability of then getting {a,c} in the second draw (which is 1/4), so the probability of getting {a,c} in this way is 1/16. Exactly equivalent is the case where you get {a,b,d} after the first draw and {a,d} after the second; that gives another 1/16. Then there's a whole other possibility to consider: If you get {a,c,d} on the first draw, then your possibilities after the second draw are {a,c}, {a,d}, {c,d}, and {a,c,d}. Out of those four, two are good: {a,c} and {a,d}. So the probability of getting a good result down this path is the probability of getting {a,c,d} on the first draw, which is 1/4, multiplied by the probability of getting either {a,c} or {a,d} on the second, which is 1/2; this gives 1/8. Then 1/16+1/16+1/8=1/4. —Bkell (talk) 03:47, 14 July 2011 (UTC)

Great response; you've interpreted my shorthand correctly. First, I know that the good cases lead to each other, that's the point I'm trying to make by saying they're "good". On paper, I draw lines connecting the cases that can lead to each other, but my means of doing this here are limited. Further, I didn't forget about the blanks, they're implied by the notes in each row. In any case, we can't exclude some of the cases merely due to the outcome of one particular dealing, so I don't understand how there couldn't be 6 possible cases at the 2nd shuffling (not counting that we might have drawn a blank and remained at the first row). Furthermore, all unique cases have been listed, even if at the 2nd withdrawal I get a blank card, since by default such a case is described in the first row and has already been counted. This last point is very significant, and I hope you realize that we can't count cases that have already been counted in our computation, which would result in inflation. I think it is a unique coincidence we got exactly the same result, but I think my method is correct. Numbility (talk) 03:58, 14 July 2011 (UTC)

Okay, here's a table of the possible outcomes of the two draws in the simplified problem. Let's say the cards we start with are A♠, 2♠, A♥, A♦.

		Second draw
		A♠	2♠	A♥	A♦	blank
First draw	A♠	(impossible)	BAD	BAD	GOOD	BAD
	2♠	BAD	(impossible)	BAD	GOOD	BAD
	A♥	BAD	BAD	(impossible)	BAD	BAD
	A♦	GOOD	GOOD	BAD	(impossible)	BAD

Out of the 16 possible outcomes, 4 of them are good. This gives a probability of 1/4.

If we call a first draw of A♥ a failure, because it cannot lead to a good outcome, and the other three possible first draws successes, then the probability of a successful first draw is 3/4; we both agree on this. Your method gives a calculation of (3/4)×(1/3). This would make sense if the probability of success on the second draw was a constant 1/3, independent of the result of the first draw (given that the first draw was a success). But you can see that this probability is not constant: The first two rows of the table above have just one good outcome each, while the third row has two good outcomes. And there are four possible outcomes in each row, so a fraction of 1/3 doesn't even make sense.

The reason conditional probability comes into play here is that, given two events A and B, the probability of both A and B occurring is P(A)×P(B|A), that is, the probability of A, multiplied by the probability of B given A. It is not simply P(A)×P(B), unless the two events are independent, in which case P(B|A)=P(B). In these problems the two events you are trying to use are not independent: The probability of being left with {a,c} after the second draw is different depending on whether or not you were left with {a,b,d} after the first draw, for example.

(By the way, you are misusing the word "unique" as it is used in mathematics. "Unique" means "only one exists"—so, for example, we may talk about "the unique solution" to an equation, if that equation has one and only one solution. So it doesn't make sense to talk about "1,800 unique cases"—if there are 1,800 of them, they are not unique! You mean "1,800 distinct cases".) —Bkell (talk) 07:01, 14 July 2011 (UTC)

Perhaps this is the core of the problem: Your multiplication for counting possibilities isn't quite right. Let's consider a simple counting problem. Suppose I choose to write either "ABC" or "CDE" on a card, and then I choose one of the letters on the card to color red. How many possibilities are there for the final result on the card?

Clearly there are six: I have two choices for the word I write on the card, and then, for each of those two choices, I have three choices for the letter I color red. Two times three is six. This is called the rule of product. We can list out the six possibilities: ABC, ABC, ABC, CDE, CDE, and CDE.

Your method of counting in the probability problems above, however, seems to proceed like this: After the first step, there are two possibilities for what I have written on the card (ABC or CDE); this is true. After the second step, there are five possibilities for the letter colored red (A, B, C, D, or E); this is also true, in a way. Therefore, this method concludes, there must be 2×5=10 possible results. Of course, you can see that this is incorrect. The reason is that some of the possibilities in the second step (coloring a letter red) are impossible for some of the possibilities in the first step (choosing ABC or CDE): If I choose ABC in the first step, then I cannot possibly choose D or E to color red in the second step. So 2×5 counts too many.

Likewise, in the merry-go-round problem, you say that after the first step there are six possibilities (GGDDD, BGDDD×2, BGGDD×3), which is correct, and then you say that after the second step there are 30 possibilities. Now, I count 21 possibilities after the second step (one way to have GGDDD remaining, two ways to have BGDDD, three ways to have BGGDD, two ways to have GDDD, three ways to have GGDD, one way to have BDDD, six ways to have BGDD, and three ways to have BGGD), but either way, we do not have all 30 or 21 or however many possibilities for each one of the six possibilities in the first step (for instance, the possibility BGDD after the second step is impossible if the outcome of the first step was GGDDD), so multiplying these numbers together is incorrect. —Bkell (talk) 07:30, 14 July 2011 (UTC)

Thanks for your detailed and thorough responses; this really helps me get to the logic of your thinking, which I'm striving to see as valid. I truly appreciate the time you're taking here to explain things. There are a few things I'd like to mention, however: in my original card problem, there are only 4 cards, namely, 2 spades, 1 heart, and 1 diamond - you mistakenly put 5 cards, so your interpretation of the problem doesn't pertain to the problem I posed; furthermore, I have been following the rule of product every step of the way, which is why I know, at the end of the day, as in the case of the card problem, I know there are only 2 ways the desired result can appear (because there are 2 spades and 1 heart), and as in the case of the merry-go-round problem, I know there are only 6 ways the desired result can appear (3 dogs, 2 girls, and 1 boy, the product of which is 6); and, yes, my knowledge of technical terms in mathematics isn't precise, and I never attempted to use "unique" in the sense that would be appropriate here, as I only expect my meaning to get across through lay-terms, which I succeeded in doing, but thanks for pointing it out to me. Let me try to illustrate my thinking in a modified version of the card problem, and we'll just go from there:

There are 2 spades and 2 hearts. After the deck has been shuffled, a card is withdrawn, then a blank is inserted in its place, and the deck is shuffled once more. What is the probability that there will be one of each card remaining after 2 shufflings and withdrawals?

a=1, b=1; c=2, d=2 (4 items, 2 of 2 kinds)
(a=b=spade; c=d=heart)
1st	(one blank	{a,b,c}*2cases	{a,c,d}*2cases			= 4cases
	for all cases)	good		good

2nd	(two blanks	{a,b}*1case	{a,c}*4cases	{c,d}*1case	= 6cases
	for all cases)	bad		good		bad

Thus, there are 4 (rule of product and addition - there is 2 of one kind, and two different cases of this, so 2+2=4)
out of 4 cases that are good at the 1st withdrawal, so the probability is 1; there are 4 (rule of product -
there are 2 of each kind, so 2*2=4) out of 6 cases that are good, so the probability is 2/3 of obtaining 1 of each card.

Wherein lies the flaw with this mode of analysis? Isn't the probability of this occurring 2/3? If not, why exactly? Point out the step(s) which are erroneous to me and I'll try to see if the underlying logic of the criticism is correct or intuitively obvious.

In anticipation of your form of analysis, I'll try to emulate it: at the first step we have 4 in 4 chances of making a desired withdrawal, so 1/1, then, at the second, we have 2 in 3 chances of making a desired withdrawal, and the two chances together make: (1/1)(2/3) = 2/3, so the chance of making the desired outcome is 2/3. Have I followed you correctly? If so, I'm not sure why my method appears appealing to me, but I need to know why it's wrong. Numbility (talk) 14:49, 14 July 2011 (UTC)

One error in your analysis is that you have omitted the cases where the second draw draws a blank and replaces it with a blank, so the final deck is {blank, S, H, H} or {blank, S, S, H}. The probability of the final deck being {blank, blank, S, H} is 1/2. Gandalf61 (talk) 15:38, 14 July 2011 (UTC)

Yes, I see that, and hadn't yet posted about that, waiting for Bkell's verdict. Without loss of generalization, I can say then about the merry-go-round is that at the first spin, I have 5/6 good cases, then the second 4/6 or 2/3 good cases, and the third 2/6 or 1/3 cases; therefore, the probability that 1 dog, 1 boy, and 1 girl will remain is: 10/54 or 5/27. Correct? Where am I failing to get the supposedly correct 1/6? Numbility (talk) 17:07, 14 July 2011 (UTC)

I do not have five cards. I have four cards: A♠, 2♠, A♥, A♦. I also have a column in my table for a blank, but only for the second draw, because your problem stated, "a card is withdrawn and a blank is inserted in its place." Please note that I have labeled some entries in my table as "(impossible)", because according to the problem as you gave it, it is impossible to draw, say, A♠ twice in a row. So the table is really a 4×4 table—one cell in each row is not actually there. —Bkell (talk) 05:42, 15 July 2011 (UTC)

I don't know how to answer your questions any more clearly. Your method does not give the correct answer for the merry-go-round problem, and that should be convincing proof that it is not valid. Several people have explained why the answer is 1/6. I went so far as to explicitly list out all 216 possible outcomes and label them as good or bad, and you can see by brute-force counting that the probability is 36/216=1/6. If you still don't believe that the probability is 1/6, I don't know how to make it clearer to you. If you won't accept this complete listing of all possible outcomes, and a count of how many of them are "good", which is basically the fundamental definition of probability, then what will you accept as proof that the answer is 1/6?

For your two-spade/two-heart problem, the answer is 1/2, not 2/3. Here is another table, listing all possible outcomes, to show that:

		Second draw
		A♠	2♠	A♥	2♥	blank
First draw	A♠	(impossible)	BAD	GOOD	GOOD	BAD
	2♠	BAD	(impossible)	GOOD	GOOD	BAD
	A♥	GOOD	GOOD	(impossible)	BAD	BAD
	2♥	GOOD	GOOD	BAD	(impossible)	BAD

Here's how to read that table. If the first draw is A♠ (corresponding to the first row), then the second draw can be either 2♠, A♥, 2♥, or blank. Draws of A♥ and 2♥ lead to a desired outcome; draws of 2♠ or blank lead to an undesired outcome. The other rows should be interpreted similarly. I do not have "five cards" here—I have five columns because, overall, there are five things you might get on the second draw (A♠, 2♠, A♥, 2♥, or blank), so I have a column for each of them. Of course, for any one given card for the first draw, only four of these five are actually possible, so one column in each row is marked "(impossible)". [If you prefer, you can remove the "blank" column and reinterpret the intersection of, say, A♠ and A♠ to mean that the second draw actually produced a blank, thereby changing all of the "(impossible)" cells to "BAD" cells.] Looking at this table, it is clear that out of the 16 (not 20) possible outcomes, 8 of them are good, so the probability is 8/16=1/2. If your analysis gives 2/3 as the answer, then the analysis cannot be valid.

Gandalf61 has already pointed out a flaw in your analysis, namely, that you apparently failed to include the possibility of getting a blank in the second draw. If you enforce a rule that the blank must not be drawn in the second draw (perhaps by not putting the blank into the deck at all), then the probability is indeed 2/3. We can see that with another table listing all possible outcomes:

		Second draw
		A♠	2♠	A♥	2♥
First draw	A♠	(impossible)	BAD	GOOD	GOOD
	2♠	BAD	(impossible)	GOOD	GOOD
	A♥	GOOD	GOOD	(impossible)	BAD
	2♥	GOOD	GOOD	BAD	(impossible)

I have removed the "blank" column because now we are not adding a blank card after the first draw. In this table, out of the 12 (not 16) possible outcomes, 8 of them are good, so the probability of a good outcome is 8/12=2/3.

If you would like me to attempt to pinpoint specific flaws in your reasoning, please explain your analysis much more thoroughly, in words rather than in the tables and symbols you've been giving, and do so for one of the problems we've discussed rather than introducing a new one. Also, give as clear and precise a justification as possible for every step of your analysis. Remember, in mathematics, reasoning is not "correct until proven wrong"—it is "incorrect until completely justified from earlier established truths." —Bkell (talk) 06:13, 15 July 2011 (UTC)

Let me repeat again an important point about the rule of product. The rule of product says that, if there are m ways to perform one task, and, for every one of those m ways, there are n ways to perform a second task, then there are m×n ways to perform the two tasks together. The phrase "for every one of those m ways" is crucial here. If some of the m ways of performing the first task correspond to more or fewer than n ways to perform the second task, then the rule of product cannot be used.

Rephrased in terms of probabilities, the rule of product says that if one experiment has probability p of success, and, for any successful outcome of the first experiment, a second experiment has probability q of success, then the probability that both experiments are successful is pq. Again, the phrase "for any successful outcome of the first experiment" is crucial here.

Let's consider your analysis of the two-spades/two-hearts problem, ignoring the failure to consider getting a blank on the second draw. (In other words, I am going to consider your analysis as written; I am not going to insert a blank card into the deck after the first draw, so that the second draw will never produce a blank.) Your analysis seems to be flawed in its use of the rule of product. Your two experiments are the first draw and the second draw. The probability of success in the first draw is, as you say, 1. However, you then appear to claim that, for any successful outcome of the first draw, there are six possibilities for the second draw. This is false. Instead, for any successful outcome of the first draw, there are only three possibilities for the second draw, as follows:

• If the first draw is A♠, then the three possibilities for the second draw are 2♠, A♥, and 2♥. Two of these (A♥ and 2♥) are good, so the probability of success in the second experiment is 2/3.
• If the first draw is 2♠, then the three possibilities for the second draw are A♠, A♥, and 2♥. Two of these (A♥ and 2♥) are good, so the probability of success in the second experiment is 2/3.
• If the first draw is A♥, then the three possibilities for the second draw are A♠, 2♠, and 2♥. Two of these (A♠ and 2♠) are good, so the probability of success in the second experiment is 2/3.
• If the first draw is 2♥, then the three possibilities for the second draw are A♠, 2♠, and A♥. Two of these (A♠ and 2♠) are good, so the probability of success in the second experiment is 2/3.

Since, for any successful outcome of the first draw, the probability of success in the second draw is 2/3 (not 4/6, even though these are numerically equal), here is where we can use the rule of product to get that the final probability is 1×(2/3)=2/3.

The flaw in your analysis seems to be that you are considering all possibilities after the second draw and including all of them with each one of the possibilities after the first draw—even pairings that don't make sense (like pairing {A♠, 2♠, A♥} after the first draw with {A♠, 2♥} after the second draw—this pairing is impossible, because you can't have 2♥ remaining after the second draw if you didn't have it remaining after the first draw). When you consider only pairings that make sense, you will see that every possibility of the first draw should have three, not six, possibilities for the second draw. —Bkell (talk) 06:58, 15 July 2011 (UTC)

I would like to apologize for misreading a part of your note: I mistakenly thought you'd considered 5 cards, which is obviously wrong. Anyway, considering everything and the overwhelming evidence, I consider this issue resolved and the answer to the original problem 1/6. Thanks! Numbility (talk) 13:16, 15 July 2011 (UTC)