Question

Christy draws 3 cards from the deck one at a time without replacement. What is the probability of selecting the cards in the order: diamond, spade and diamond?

Answer 1

Start with a deck of 52 cards. Draw 1 card. Probability of drawing a diamond first:

$\dfrac{\binom{13}1}{\binom{52}1}$

Now draw another single card from the deck, which now contains 51 cards. Probability of drawing a spade:

$\dfrac{\binom{13}1}{\binom{51}1}$

Draw 1 last card. The deck has 50 cards left, and 12 of these are diamonds. Probability of drawing a diamond:

$\dfrac{\binom{12}1}{\binom{50}1}$

So the probability of drawing a diamond, then a spade, then another diamond is

$\dfrac{\binom{13}1}{\binom{52}1}\cdot\dfrac{\binom{13}1}{\binom{51}1}\cdot\dfrac{\binom{12}1}{\binom{50}1}=\dfrac{13}{850}$