If you draw two cards at random without replacement from a s

If you draw two cards at random (without replacement) from a standard deck of 52 playing cards, what is the probability that:

(a) One will be red and one will be black?

(b) Both cards are spades?

Solution

a) There are 26 red cards and 26 black cards in a standard deck of 52 playing cards.

Total number of ways of selecting 2 cards will be : ⁵²C 2 = 1326 ways

The number of ways in which one red out of 26 red cards will be chosen = ²⁶ C ₁

The number of ways in which one black out of 26 black cards will be chosen = ²⁶ C ₁  

So total number of ways =²⁶ C _{1 *} ²⁶ C ₁ =676

Therefore the probability that one will be red and one will be black = 676/ 1326 = 169/334 (ans )

b) Both cards are spades:

Out of 52 cards,there are 13 spades.. So probability of drawing a spade on first attempt will be 13/52
After the first attempt, there will be 12 spades left out of 51 total. Since the cards are not being replaced,the probability of drawing spade on second attempt will be 12/51.
Therefore the probability that both cards are spades will be :(13/52) * (12/51) = 156/2652 = 1/17 (ans )