Balls of Two Colors from Interactive Mathematics Miscellany and Puzzles

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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 W_i and B_i 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(W_i) = w/(w + b), for any i not exceeding the total number of balls w + b.

Solution

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

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

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