Probability your third card is a face card

Question

A standard 52-card deck is shuffled uniformly and you are dealt five cards, one at a time. What is the probability that your third card is a face card (jack, queen, or king)?

Accepted Answer

By symmetry, the third card dealt is equally likely to be any of the 52 cards. There are 12 face cards (3 ranks in each of 4 suits), so P(	ext{third card is a face card}) = \frac{12}{52} = \frac{3}{13} \approx 0.2308. This is unaffected by your first two cards as long as you have not looked at them. Unconditionally, position three is exchangeable with every other position, so it inherits the deck-wide rate of face cards, 12/52. If you had seen the first two and both were face cards, you would update: 10 face cards left among 50 unseen, P = 10/50 = 1/5. But the plain question, with nothing re…

Probability your third card is a face card

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