You are dealt a five card hand at random which consists of 3

You are dealt a five card hand at random which consists of 3 aces and 2 other cards from a standard 52-card deck (recall such decks have 4 aces total). You now get to exchange cards. You decide to exchange the 2 non aces in your hand for 2 new cards (dealt at random from the remaining deck). What is the probability you end up with a four of a kind in your new hand?

Solution

If you want a four of a kind, then one of the 2 new cards must be the remaining ace.

There are 47 other cards in the deck. There are 47C2 ways to choose any two cards there.

There is one way to get the last ace, and 46 ways to get its partner. Thus, there are 46 ways to deal a four of a kind.

Thus,

P(four of a kind) = 46/(47C2) = 0.042553191 [ANSWER]