Card Probability

Question

Find the probability of drawing an ace and a queen when two cards are drawn without replacement from a standard deck of cards.

Answer 1

There are four aces and four queens in a standard deck of 52 cards. Aces and queens are separate ranks and thus independent of each other, so no shenanigans with drawing an ace and a queen at the same time.

As the order apparently does not matter, we must then consider the two possible outcomes for a two-card draw that match the desired outcome: we can first draw and ace, then draw a queen; or we can first draw a queen, then draw an ace. We'll call these Outcome A and Outcome B, respectively.

Outcome A: First we draw one of the four aces out of all 52 cards, which has a probability of 4/52. As the ace is not replaced, we have 51 cards remaining. We then draw one of the four queens out of the 51 remaining cards, which has a probability of 4/51. As drawing the queen depends on first drawing the ace, we will multiply the two probabilities to get the probability of Outcome A: 4/52 * 4/51 = 16/2652 ≈ 0.00603

Outcome B: The probability for Outcome B is the same as Outcome A. We're just drawing the two cards in a different order. The probability will still be 4/52 * 4/51 = 16/2652 ≈ 0.00603.

We can then add the probabilities of the two outcomes together to get 16/2652 + 16/2652 = 32/2652 ≈ 0.01206.

Answer 2

There are 4 aces and 4 queens in a 52 card deck.

Event A: Probability of drawing an ace then a queen = 4/52 * 3/51 = 4/663

Event B: Probability of drawing a queen then an ace = 4/52 * 3/51 = 4/663

Probability of Event A or Event B. Since the events are mutually exclusive, the probability equals probability of event A + probability of event B = 8/663