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So here is the question:

There is a bag of marbles with an equal number of black and red marbles in it. I randomly draw marbles. I will stop drawing when I have at least one black marble and at least one red marble. What is the expected number of marbles that I draw?

I couldn’t find which probability distribution to use to solve this question. First, I thought that this is a geometric but without replacement. Then, I thought that this is a hypergeometric but i couldn’t find the answer anyway. Help me to solve this question,

Thanks.

Efe Gürtunca
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  • Are you drawing with or without replacement? – lulu Nov 25 '18 at 15:08
  • Assuming you meant "without replacement", a good way to get started is to work it explicitly for small collections. That should give you some ideas about a general solution and it will give you several solid values that you can test your eventual expression against. – lulu Nov 25 '18 at 15:10
  • I tried to draw without replacement and as you said I found the expected value when there are 4 marbles and 6 marbles.(when 4 marbles expected value=7/3 and when 6 marbles expected value=25/10). Then, I tried to find a general expression for the solution but I couldn't. – Efe Gürtunca Nov 25 '18 at 15:14
  • Well, assuming you start with $N$ of each, what's the probability of drawing $i$ red followed by black? – lulu Nov 25 '18 at 15:14
  • To make it look more like a standard hypergeometric problem, note that the first draw doesn't really matter. It's either red or black, and in either case you are down to a standard hypergeometric calculation. That should simplify things. – lulu Nov 25 '18 at 15:17
  • I don't understand how it is going to be a hypergeometric problem. I found a formula which is P(N=n)=(combin(X, n-1))/(combin((2x-1)/(n-1))). In that formula, I assume that I've picked 1 red and now there are x-1 red and x black. But I couldn't find the expected value of this formula. I would be very glad if you could help... – Efe Gürtunca Nov 25 '18 at 16:10
  • In this formula, for example, if i've picked the black marble in the first choose n=1, in the second choose n=2 and so on – Efe Gürtunca Nov 25 '18 at 16:11
  • See, e.g., this question as a near duplicate. – lulu Nov 25 '18 at 18:59

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