Fifteen guests drawing raffle numbers

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At a fair, 15 guests each draw a raffle number independently and uniformly from 1 to 100 (numbers can repeat, since each draw is independent). What is the probability that at least two guests draw the same number?

Accepted Answer

Rather than count all the ways a duplicate can appear, compute the chance all fifteen numbers are distinct and subtract from 1. Draw the numbers one guest at a time, each needing a value not already drawn: - Guest 1: probability 100/100. - Guest 2: avoids 1 used number, probability 99/100. - Guest 3: avoids 2 used numbers, probability 98/100. - down to guest 15, which avoids 14 used numbers, probability 86/100. Multiplying all fifteen factors: P(	ext{all distinct}) = \frac{100}{100}\cdot\frac{99}{100}\cdots\frac{86}{100} \approx 0.331. Therefore P(	ext{at least two match}) = 1 - 0.331 \appr…

Fifteen guests drawing raffle numbers

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