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 the probability of choosing a black card first

There are \(13 + 13=26\) black cards in a deck of \(52\) cards. The probability of choosing a black card \(P(B)=\frac{26}{52}=\frac{1}{2}\)

Step2: Calculate the probability of choosing a heart second

There are \(13\) heart cards in a deck of \(52\) cards. Since the card is replaced, the probability of choosing a heart \(P(H)=\frac{13}{52}=\frac{1}{4}\)

Step3: Calculate the combined probability

Since the two events (choosing a black card first and a heart second) are independent (because of replacement), we use the formula \(P(A\cap B)=P(A)\times P(B)\). So \(P = \frac{1}{2}\times\frac{1}{4}=\frac{1}{8}\)

Answer:

\(\frac{1}{8}\) (corresponding to the first option)