Not the last piece of the puzzle

Question

A game awards one of 6 equally likely medals per match, independent each time. You play until you have earned all six. One medal is bronze.

What is the probability that bronze is NOT the last new medal you earn?

Accepted Answer

Every medal is awarded with the same probability each match, so the six medals are interchangeable. By symmetry, each is equally likely to be the last new one, with probability \frac{1}{6}. The question asks for the complement, "bronze is not last," so subtract from 1: P(	ext{bronze not last}) = 1 - \frac{1}{6} = \frac{5}{6} \approx 0.8333. Equivalently, any of the other 5 medals is last, giving 5 	imes \frac{1}{6} = \frac{5}{6}. Takeaway. Each of n equally likely types is last with probability \frac{1}{n}, so a named type is not last with probability \frac{n-1}{n}; here \frac{5}{6}.

Not the last piece of the puzzle

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