What is the probability that a player is dealt 3 kings in a single euchre hand?

Question

In the game of euchre, the deck consists of the nine, ten, jack, queen, king and ace of each suit. Players are dealt a five card hand.

What is the probability that a player is dealt 3 kings in that hand of five cards?

Answer 1

This is a hypergeometric distribution problem. Luckily, calculators exist that let you just plug in the numbers and get an answer. I used this one with

Population size = 24 (the size of the deck)

Number of successes in population = 4 (the total number of kings)

Sample size = 5 (the number of cards in your hand)

Number of successes in sample = 3 (the number of kings you are interested in)

The results show that the chances of getting exactly 3 kings is 1.79%. If you also include hands with 4 kings, it goes up to 1.84%.

Answer 2

You could also think of it like this, using combinations. Recall that a combination C(x, n) tells you the number of ways you could arrange n objects from a set of size x where order doesn't matter. So if you have 10 books, C(10, 3) tells you the number of ways you could select 3 of them.

There are 4 Kings in a Euchre deck, and 24 total cards. So there are C(4, 3) = 4 ways of choosing 3. Now since you need 2 other cards to finish out your hand, and you have already chosen from the 4 kings to leave 20 remaining cards, there are C(20, 2) = 190 ways to choose the remaining cards.

So there are 4 * 190 ways to form a hand of 3 Kings (and 2 other cards). There are C(24, 5) = 42504 total possible Euchre hands. So your probability is:

P = (4 * 190) / 42504 = 780 / 42504 = .01788 ~= 1.788% chance.