Questions: 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? 1/8 1/2 2/3 3/4

Solution Steps

A standard deck of cards has 52 cards in total. There are 26 black cards (13 spades and 13 clubs). Since the card is replaced after each pick, the probability of drawing a black card first is:

\[ P(\text{Black card first}) = \frac{26}{52} = \frac{1}{2} \]

Since the card is replaced, the deck remains full with 52 cards. There are 13 hearts in the deck. Therefore, the probability of drawing a heart second is:

\[ P(\text{Heart second}) = \frac{13}{52} = \frac{1}{4} \]

The events are independent because the card is replaced after each draw. Therefore, the probability of both events occurring (drawing a black card first and a heart second) is the product of their individual probabilities:

\[ P(\text{Black card first and Heart second}) = P(\text{Black card first}) \times P(\text{Heart second}) = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} \]

Final Answer