Balls of Two Colors

An urn contains w white balls and b black balls (w > 0 and b > 0). The balls are thoroughly mixed and two are drawn, one after the other, without replacement. Let Wi and Bi denote the respective outcomes 'white on the ith draw' and 'black on the ith draw,' for i = 1, 2.

Prove that P(W2) = P(W1) = w/(w + b). (Which clearly implies a similar identity for B2 and B1.)

Furthermore, P(Wi) = w/(w + b), for any i not exceeding the total number of balls w + b.

Solution

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Copyright © 1996-2012 Alexander Bogomolny

We must remember the formula for the total probability:

P(W2) = P(W2|W1)·P(W1) + P(W2|B1)·P(B1),

from which

P(W2) = (w - 1)/(b + w - 1)·w/(b + w) + w/(b + w - 1)·b/(b + w) =
w/(b + w - 1)(b + w)·(w - 1 + b),

which is simplified to w/(w + b).

References

  1. Ruma Falk, Understanding Probability and Statistics, A K Peters, 1993

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Copyright © 1996-2012 Alexander Bogomolny

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