A card drawn in both of two batches

Question

From a 52-card deck you draw a first batch of 8 cards with replacement, recording them. Independently, you draw a second batch of 8 cards with replacement. What is the probability that at least one card value appears in both batches?

Accepted Answer

This is a cross collision: we look for a card that shows up in batch one and also in batch two, ignoring repeats within a single batch. Count the cross pairs, one card from each batch: 8 	imes 8 = 64 	ext{ cross pairs}. Each cross pair matches with probability 1/52. The chance of no shared card is approximately P(	ext{no shared card}) \approx e^{-64/52} = e^{-1.231} \approx 0.292. Therefore P(	ext{at least one shared card}) \approx 1 - 0.292 \approx 0.708, about 71\%. The expected number of matching cross pairs is 64/52 \approx 1.23, a bit above 1, so an overlap is the likely outcome. Tak…

A card drawn in both of two batches

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