Probability Question

Question

In poker, a 5-card hand is a flush when all five cards are of the same suit (hearts, diamonds, spades, or clubs). Note a standard deck of cards has 52 cards of 4 suits with each suit having cards numbered 2 through 10, a jack, a queen, a king, and an ace. Let 𝐴 be the event of getting dealt a flush. Let 𝐵 be the event of getting dealt a hand in which at least two cards have the same face value (2 thru 10, jack, queen, king, ace).

True or False: 𝐴 and 𝐵 are dependent event.

Answer 1

If you were dealt a flush and another player were dealt 2 of a kind, from the same deck, without replacement, the A and B would be dependent events. Knowing you had a flush would alter the probability the other player had 2 of a kind. There would be less ways to get 2 of a kind, if 5 cards were removed from the deck of the same suit.

But if the cards were replaced, and the 2 of a kind were dealt from a 52 card deck, then getting a flush from one 52 card deck will not change the probability that a 2nd hand from a full 52 card deck will be 2 of a kind.