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?
\\(\frac{1}{8}\\)
\\(\frac{1}{2}\\)
\\(\frac{2}{3}\\)
\\(\frac{3}{4}\\)

Explanation:

Step1: Calculate probability of choosing a black card

There are 26 black cards in a 52 - card deck. So the probability of choosing a black card $P(B)=\frac{26}{52}=\frac{1}{2}$.

Step2: Calculate probability of choosing a heart

There are 13 hearts in a 52 - card deck. So the probability of choosing a heart $P(H)=\frac{13}{52}=\frac{1}{4}$.

Step3: Calculate the combined probability

Since the two events are independent (because the card is replaced), the probability of choosing a black card first and a heart second is $P = P(B)\times P(H)$. Substitute the values: $P=\frac{1}{2}\times\frac{1}{4}=\frac{1}{8}$.

Answer:

A. $\frac{1}{8}$