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 first

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

Step2: Calculate probability of choosing a heart second

Total cards = \(52\). Hearts = \(13\).
Probability \(P(\text{heart})=\frac{13}{52}=\frac{1}{4}\)

Step3: Use multiplication rule for independent events

Since the cards are replaced, the two events are independent.
\(P(\text{black then heart})=P(\text{black})\times P(\text{heart})\)
\(P=\frac{1}{2}\times\frac{1}{4}=\frac{1}{8}\)

Answer:

\(\frac{1}{8}\)