in a standard deck of cards there are 13 spades, 13 clubs, 13 hearts, and 13 diamonds. the spades and the…

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?\n$\frac{1}{8}$\n$\frac{1}{2}$\n$\frac{2}{3}$\n$\frac{3}{4}$

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?\n$\frac{1}{8}$\n$\frac{1}{2}$\n$\frac{2}{3}$\n$\frac{3}{4}$

Answer

Explanation:

Step1: Calculate probability of choosing black card

There are 26 black cards (13 spades + 13 clubs) in a 52 - card deck. The probability of choosing a black card on the first draw is $P(\text{black})=\frac{26}{52}=\frac{1}{2}$.

Step2: Calculate probability of choosing a heart

There are 13 hearts in a 52 - card deck. The probability of choosing a heart on the second draw is $P(\text{heart})=\frac{13}{52}=\frac{1}{4}$.

Step3: Use multiplication rule for independent events

Since the draws are independent (cards are replaced), the probability of both events occurring is the product of their individual probabilities. So $P = P(\text{black})\times P(\text{heart})=\frac{1}{2}\times\frac{1}{4}=\frac{1}{8}$.

Answer:

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