A poker hand consisting of 5 cards is dealt from a standard deck of 52 cards. How do you find the probability that the hand contains exactly 3 face cards.

Answer 1

Hi Ava! For this question, let's review some basic facts about a deck of cards. There are 52 cards in a deck, split evenly among 4 suits. This means each suit has 13 cards: 10 number cards (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10) and three face cards (Jack, Queen, King). This means that there are (3 face cards per suit)(4 suits per deck) = 12 face cards per deck.

So we know that the probability of pulling one face card must be 12/52, and the probability of pulling one number card are 40/52. We know also that we're drawing a full hand, so we must not be replacing cards. In other words, our deck of cards will decrease in size by one card each time we draw. We should reflect this in a fraction: the probability of drawing a face card first is 12/52, but now that there are 11 face cards left and 51 cards left, the chance we draw a second face card is 11/51. We can repeat this for the other two face cards in our hand: the odds of pulling three face cards is (12/52)(11/51)(10/50).

The method we use to figure out our odds of pulling a number card are similar: we want our last two cards to be number cards, of which there are still 40 left. But we have already pulled 3 cards from our deck, so the remaining cards total 49. Therefore, the probability of our fourth card being a number card is 40/49, and we symbolically remove a card from the top and bottom of that fraction to see that the probability of our fifth card being a number card are 39/48. Finally, we multiply all of our fractions together to represent the probability that these independent events happen at the same time. Our final answer for the probability of having 3 face cards and 2 number cards is (12/52)(11/51)(10/50)(40/49)(39/48) = 0.00660264, or 0.66203%.