In a standard deck, two cards are drawn without replacement. What is the probability that both cards are aces?

• A. 1/169
• B. 1/132
• C. 1/221
• D. 1/441

Answer: (C)

When drawing two cards without replacement, the second draw depends on what happened the first time. To get both cards as aces, multiply the chance the first card is an ace by the chance the second is an ace given the first was an ace: 4/52 for the first ace, and then 3/51 for the second ace. So the probability is (4/52) × (3/51) = (1/13) × (1/17) = 1/221.

You can also see this by counting: the number of ways to pick two aces is C(4,2) = 6, and the total number of two-card hands is C(52,2) = 1326, so 6/1326 = 1/221. The other options come from treating the draws as if they were independent or miscounting the remaining aces after the first draw, which isn’t correct for drawing without replacement.