# A standard deck of playing cards consists of 52 cards. Each card has a rank and a suit.

A standard deck of playing cards consists of 52 cards. Each card has a rank and a suit. There are 4 possible suits (spades, clubs, hearts, diamonds) with 13 cards each. Assume that the deck is perfectly shuffled (that is, all outcomes are equally likely).

What is the probability that a hand of 5 cards dealt from the deck contains only diamonds given that the first 3 cards in the hand are the ace of diamonds, the queen of diamonds, and the king of diamonds?

$n|S|=\binom{52}{5}$

$n|E|=4*\binom{13}{3}*\binom{48}{2}$

=> $\frac{4*\binom{13}{3}*\binom{48}{2}}{\binom{52}{5}}$

• Your answer is incorrect. The favorable cases consists of hands that include five diamonds. Since you are told that the hand includes the $\color{red}{A\diamondsuit}, \color{red}{Q\diamondsuit}, \color{red}{K\diamondsuit}$, you need two more diamonds. – N. F. Taussig May 5 '18 at 20:13
There are 49 cards left, of which 10 are diamonds. The probability that the next card is a diamond is $\frac{10}{49}$ and probability that the next card after is also a diamond is $\frac{9}{48}$. Therefore what you want is $\frac{10\times 9}{49\times 48}$.