In a 5-card hand

In a 5-card hand, what is the probability of getting four face cards?

It is a 52-card deck, and it has 12 face cards. That means there are 40 non-face cards. The required combination should include five cards, in which four of which are going to be face cards and one of them is going to be a non-face card.

Since the order in which they come doesn’t matter, we use combinations. So the answer is

Out of the 12 face cards, we choose four and out of the 40 other cards, we choose 1, divided by all possible combinations, i.e. out of the 52 cards, choose 5.

₁₂C₄ x ₄₀C₁ / ₅₂C₅ = 0.00076

Three cards are kings

Out of the 4 kings, we choose three kings and out of the 48 other cards, we choose 1 non-king

P = ₄C₃ x ₄₈C₂ / ₅₂C₅ = 0.001736

All five cards are hearts

P = ₁₃C₅ / ₅₂C₅ = 0.000495