Two raffle groups sharing a ticket number

Question

Group A draws 15 raffle tickets and group B draws 10, each ticket an independent uniform number from 1 to 200 (numbers can repeat). What is the probability that at least one ticket number appears in both groups?

Accepted Answer

This is a cross collision: we want a group-A number that also shows up in group B, ignoring repeats inside a single group. Count the cross pairs: 15 	imes 10 = 150 	ext{ cross pairs}. Each cross pair matches with probability 1/200. The chance of no shared number is approximately P(	ext{no shared number}) \approx e^{-150/200} = e^{-0.75} \approx 0.472. Therefore P(	ext{at least one shared number}) \approx 1 - 0.472 \approx 0.528, about 53\%. The expected number of matching cross pairs is 150/200 = 0.75, just under 1, which is why the probability lands close to a coin flip. Takeaway. Cross …

Two raffle groups sharing a ticket number

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