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Maths quiz
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Mikeyj
Unfictologist


Joined: 18 Oct 2004
Posts: 1847
Location: London

 		Maths quiz
Rubbish at probability
I really can't do these things and I suddenly thought "I know people who can" sorry to pollute the board with this nonsense, but it might be a moments puzzle fun for someone.

If we have two bags of 85 balls numbered 1-85, and one person selects 12 at random and the other selects 6 at random from the second bag, what is the probability that the second person draws two balls the same as the first?
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Irrelevant musings.

PostPosted: Thu Jul 05, 2007 11:02 am
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urthstripe
Entrenched


Joined: 31 Aug 2004
Posts: 1113
Location: Atlanta, GA

 		Is the first guy putting the balls back in after he picks them? Does he pick 12 and put the twelve back in or does he pick one and put it back in and take another one, etc?
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PostPosted: Thu Jul 05, 2007 2:19 pm
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Agent Lex
Entrenched


Joined: 11 May 2006
Posts: 1188
Location: No longer London, still in England

 		I'm going to assume that both people have separate bags with identical contents, and that neither replaces any balls during the exercise. There's 2 possibilities to work out here:

1) The probability of exactly 2 balls being the same (spoiler'd for those who want to have a go themselves)
Spoiler (Rollover to View):
It doesn't matter at all which 12 balls the first person picks, only that he has picked 12 different balls. The second person, picking six balls, can pick exactly two balls in a number of combinations, eg:
same, same, different, different, different, different;
same, different, same, different, different, different;
same, different, different, same, different, different;
etc etc. All in all, there are 5 + 4 + 3 + 2 + 1 = 15 combinations he can pick to get exactly 2 balls the same. Each combination, by the miracle of fractions, has exactly the same probability of occuring. Using the first example:
First ball same, probability 12/85
Second ball same, probability 11/84
Third ball different, probability 73/83
Fourth ball different, probability 72/82
Fifth ball different, probability 71/81
Sixth ball different, probability 70/80
Probability of all events occuring is all of those fractions multiplied, which is:
12 * 11 * 73 * 72 * 71 * 70
--------------------------------
85 * 84 * 83 * 82 * 81 * 80
Which works out to 3448146240/314894563200. Big numbers! We can reduce it down to 57013/5206590. Since there's 15 chances of this happening, we need to multiply this fraction by 15 to get the chance of exactly 2 balls being the same. The result is 855195/5206590 (this is why we reduced it), which =~0.16425


2) The probability of at least two balls being the same:
Spoiler (Rollover to View):
The easiest way to work this out is to work out the probabilities of 0 and 1 balls being the same, and subtract this from 1. The probability of 0 balls being the same is easy, 73/85*72/84*71/83*70/82*69/81*68/70 = 122565925440/314894563200.
The probability of 1 ball being the same is 6*12/85*73/84*72/83*71/82*70/81*69/80 (6 combinations in the form ABBBBB etc) = 6*21629280960/314894563200 = 129775685760/314894563200.

Since the denominator is the same, we can add these two fractions trivially (this is why we didn't reduce it). The total is 252341611200/314894563200. This is the probability of getting 0 or 1 ball the same. The probability of getting 2 or more balls is everything else, so 1 - that = 62552952000/314894563200 =~ 0.19865. Not too much more, then!


PostPosted: Thu Jul 05, 2007 3:36 pm
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Mikeyj
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Joined: 18 Oct 2004
Posts: 1847
Location: London

 		Thanks guys - Urthstripe, I don't know whether they replace balls, I think the question has been lazily phrased.

Agent Lex - thank you Smile I shall pass on your reply verbatim with a proud, but slightly glazed expression. You did multiplying and dividing and I am in awe. Smile
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PostPosted: Fri Jul 06, 2007 8:18 am
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Agent Lex
Entrenched


Joined: 11 May 2006
Posts: 1188
Location: No longer London, still in England

 		I hope I didn't just do your homework for you! Razz

The most tiring bit was probably the most unnecessary, reducing that huge fraction. But I fancied doing it anyway, since I was in the swing of it.

PostPosted: Fri Jul 06, 2007 8:24 am
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Mikeyj
Unfictologist


Joined: 18 Oct 2004
Posts: 1847
Location: London

 		Not homework no - a bit beyond that now, I have a job and everything Wink . A friend of a friend has some cell lines with different genetic features and they've asked anyone they can what the results of a particular sampling would be (only just found this out - I'm now sure they've asked the wrong questions - not my problem Smile )
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PostPosted: Fri Jul 06, 2007 8:35 am
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