Question

In a standard deck of cards there are 13 spades, 13 clubs, 13 hearts, and 13 diamonds. The spades and the clubs are black and the hearts and the diamonds are red.

If two cards are chosen at random from a deck, one at a time, and replaced after each pick, what is the probability that a black card is chosen first and a heart is chosen second?

Answer 1

There are 13 heart cards

Now 1 heart card can be drawn from a pack of 13 heart cards in 13C1 ways
13C1=13

Also,1 card can be drawn from a pack of 52 cards in 52C1 ways
52C1=52

The probability of drawing a heart card from a pack of 52 cards is 
13/52=1/4

there are 26 black cards in total 
(13 spades +13 clubs)

Now 1 black card can be drawn from a pack of 26 black cards in 26C1 ways
26C1=26

Also,1 card can be drawn from a pack of 52 cards in 52C1 ways
52C1=52

The probability of drawing a black card from a pack of 52 cards is 
26/52=1/2

Now as both the task are performed one after the other and after replacement so the probability of drawing a heart after a black card is
=1/2    *   1/4
=1/8

Answer 2

Total number of cards in a deck = 52

Number of cards that are black = 26

Number of cards that are hearts = 13

With replacement,

P(black card, then red card) = (26/52) x (13/52) = 1/8

Answer: The probability is 1/8