In a standard deck of cards, there are 13 spades, 13 clubs, 13 hearts, and 13 diamonds. The spades and clubs are black, and the hearts and 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?

A$\frac{1}{8}$
B$\frac{1}{2}$
C$\frac{2}{3}$
D$\frac{3}{4}$

Question

03-11-2026
Mathematics

Answered

In a standard deck of cards, there are 13 spades, 13 clubs, 13 hearts, and 13 diamonds. The spades and clubs are black, and the hearts and 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?

A$\frac{1}{8}$
B$\frac{1}{2}$
C$\frac{2}{3}$
D$\frac{3}{4}$

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T18:38:43-05:00

The probability of picking a black card (26 out of 52) is $\frac{1}{2}$.
The probability of picking a heart (13 out of 52) is $\frac{1}{4}$.
Since the events are independent due to replacement, multiply the probabilities: $\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$.
The final answer is $\boxed{\frac{1}{8}}$.

Explanation

Analyze the deck composition and the nature of the draws

We are dealing with a standard deck of $52$ cards. The deck consists of four suits: spades, clubs, hearts, and diamonds, each containing $13$ cards. The black cards are the spades and clubs, totaling $13 + 13 = 26$ black cards. The red cards are the hearts and diamonds, totaling $13 + 13 = 26$ red cards. We are drawing two cards sequentially with replacement, meaning the first card is put back into the deck before the second card is drawn. This ensures that the two draws are independent events.

Calculate the probability of drawing a black card first

The first event is choosing a black card. The total number of cards in the deck is $52$, and the number of black cards is $26$. Thus, the probability of drawing a black card first, denoted as $P(\text{Black}_1)$, is:
\[P(\text{Black}_1) = \frac{26}{52} = \frac{1}{2}\]

Calculate the probability of drawing a heart second

The second event is choosing a heart. Since the first card was replaced, the deck composition remains unchanged, and the total number of cards is still $52$. There are $13$ hearts in the deck. Thus, the probability of drawing a heart on the second draw, denoted as $P(\text{Heart}_2)$, is:
\[P(\text{Heart}_2) = \frac{13}{52} = \frac{1}{4}\]

Calculate the joint probability

Because the two draws are independent, the joint probability of drawing a black card first and a heart second is the product of their individual probabilities:
\[P(\text{Black}_1 \cap \text{Heart}_2) = P(\text{Black}_1) \times P(\text{Heart}_2)\]
\[P(\text{Black}_1 \cap \text{Heart}_2) = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}\]
Comparing this result to the given options, we find that the probability matches the first option.

Examples

This type of probability calculation is essential for understanding games of chance or statistical sampling with replacement. For instance, if you are drawing survey respondents or picking items from a batch where each item is returned after analysis, you can calculate the likelihood of specific sequences occurring using the same principle of multiplying independent probabilities. For example, if you have a bag with 10 marbles, 5 red and 5 blue, the probability of picking a red then a blue with replacement is $\frac{5}{10} \times \frac{5}{10} = \frac{1}{4}$.