Should you draw first or second?

Question

A bag contains k black balls and 1 red ball. Two players take turns drawing one ball at a time, without replacement. The player who draws the red ball wins. Is it better to draw first or second? Give the exact winning probability for the first player in terms of k.

Accepted Answer

If the total number of balls, k + 1, is odd, the first player has the edge; if it is even, the chances are exactly equal. Lay the balls out. Drawing without replacement is the same as shuffling the k + 1 balls into a random order and revealing them one by one. The red ball is equally likely to be in any of the k + 1 positions. The first player receives positions 1, 3, 5, ...; the second player positions 2, 4, 6, .... Count the positions. Among k + 1 positions, the number of odd positions is (k + 1)/2 rounded up. So - if k + 1 is even, each player has (k + 1)/2 positions and wins with probabil…

Should you draw first or second?

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