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: Find P(black card first)

Total cards = 52, black cards = 13 + 13 = 26.
$P(\text{black}) = \frac{26}{52} = \frac{1}{2}$

Step2: Find P(heart second)

Hearts = 13, card is replaced so total cards remain 52.
$P(\text{heart}) = \frac{13}{52} = \frac{1}{4}$

Step3: Multiply the two probabilities

Since picks are independent, multiply the probabilities.
$P(\text{black then heart}) = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$

Answer:

$\boldsymbol{\frac{1}{8}}$ (Option A. $\frac{1}{8}$)