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)

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**