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Suppose we put eight white and two black balls into a bag and then draw forth the balls one at a time.
If we repeat this experiment many times, which draw is most likely to produce the first black ball?

Try to guess (=estimate guided by intuition), then evaluate.
You'd be surprised.

Source: A.E. Lawrence, “Playing With Probability”
Mathematical Gazette, (1969)

 No Solution Yet Submitted by Ady TZIDON No Rating

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 Intuition | Comment 4 of 6 |
I thought it was pretty obvious that the 1st draw is the one most likely be the first black ball.  While each draw has a 20% probability of being black, the first one is the only one where there is no possibility of a preceding black.  And the later the draw, the greater the likelihood that there was a preceding black.  Until the 10th draw has no chance of being the 1st black.

Or, a slightly less obvious way of reaching that conclusion, still without a calculation:
Let P(n) = prob that the nth draw is black.
Let P1(n) = probability that the nth draw is the 1st black.
Let P2(n) = probability that the nth draw is the 2nd black.
Necessarily, P(n) = P1(n) + P2(n) = .2 (a constant)
The smallest P2 is P2(1) = 0.  Which means that the largest P1 must be P1(1).  All other P1(n) = .2 - P2(n) < .2

Edited on September 5, 2015, 8:33 am
 Posted by Steve Herman on 2015-09-04 17:31:21

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