# French playing cards, probability to have a four (of a kind)

We have a "French playing cards" composed of $52$ cards. Each kind of cards (Ace, King, Queen, ...) is composed of $4$ cards : There are $4$ Ace, $4$ Jack, etc. We pull $5$ cards among the $52$ cards.

We are a four (of a kind) if among the $5$ cards pulled, there are the $4$ of a same value ($4$ cards Ace, or King, or 2, or 3......).

What is the probability to have a four (of a kind) if the first card pulled is an Ace, and the fifth card pulled is a King ?

I hope you understand, it's very difficult for me to translate that. Thank you for help in advance.

• In English, As is Ace, Valet is Jack or Knave, Dame is Queen and Roi is King. Jan 25, 2018 at 20:20
• Thank you, I edit immediately ! Jan 25, 2018 at 20:25
• Some more English terminology: For the four of hearts (quatre coeurs?), the four is called the rank of the card, and hearts is called the suit of the card.
– John
Jan 25, 2018 at 20:27

Given that you have drawn one ace and one king, you will need either to draw the remaining three aces, or draw the remaining three kings, from the remaining $50$ cards.
There are two winning ways to draw the remaining three cards, and ${50 \choose 3}$ total ways to draw the remaining three cards, so the probability is $2/{50 \choose 3}$.