Let's break down the problem step by step to find the probability that a black card is chosen first and a heart is chosen second when two cards are chosen at random from a deck, one at a time, and replaced after each pick.
### Step 1: Total Number of Cards
A standard deck of cards has:
- 13 spades (black)
- 13 clubs (black)
- 13 hearts (red)
- 13 diamonds (red)
This totals to 52 cards in the deck.
### Step 2: Probability of Drawing a Black Card First
The deck has 13 spades and 13 clubs, making a total of 26 black cards. The probability of drawing a black card first is the ratio of black cards to the total number of cards.
[tex]\[
\text{Probability (Black Card First)} = \frac{26}{52} = \frac{1}{2}
\][/tex]
### Step 3: Probability of Drawing a Heart Second
Since the cards are replaced after each pick, the composition of the deck remains the same. There are 13 hearts in the deck. The probability of drawing a heart second is the ratio of hearts to the total number of cards.
[tex]\[
\text{Probability (Heart Second)} = \frac{13}{52} = \frac{1}{4}
\][/tex]
### Step 4: Combined Probability
Since the two events (drawing a black card first and drawing a heart second) are independent (because of replacement), we multiply the probabilities of each event to get the combined probability:
[tex]\[
\text{Combined Probability} = \text{Probability (Black Card First)} \times \text{Probability (Heart Second)}
\][/tex]
[tex]\[
\text{Combined Probability} = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}
\][/tex]
Thus, the probability that a black card is chosen first and a heart is chosen second is:
[tex]\[
\boxed{\frac{1}{8}}
\][/tex]
