The eighth card is the first repeat

Question

You draw cards from a 52-card deck with replacement (each draw is uniform over all 52 cards, then the card is returned). What is the probability that the eighth draw is the first to repeat a card seen earlier? (The first seven draws are all distinct, and the eighth matches one of them.)

Accepted Answer

Two parts: the first seven draws are all distinct, then the eighth matches one of them. Part 1, the first seven all distinct. Each new draw must avoid the cards already seen: P(	ext{first 7 distinct}) = \frac{52}{52}\cdot\frac{51}{52}\cdots\frac{46}{52} \approx 0.6559. Part 2, given seven distinct cards are out, the eighth draw matches one of those seven: P(	ext{8th matches} \mid 	ext{7 distinct}) = \frac{7}{52} \approx 0.1346. Multiply: P(	ext{first repeat on draw 8}) \approx 0.6559 	imes 0.1346 \approx 0.0883, about 8.8\%. Takeaway. The first repeat lands exactly on draw k with probabi…

The eighth card is the first repeat

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