To determine the probability of drawing a black card first and then a heart second from a standard 52-card deck, where each card is replaced after it is drawn, we follow these steps:
1. Determine the total number of black cards and hearts:
- In a standard deck of 52 cards, there are 26 black cards (13 spades + 13 clubs).
- There are 13 hearts.
2. Calculate the probability of drawing a black card first:
- The probability of drawing a black card from the deck is given by the number of black cards divided by the total number of cards.
- [tex]\[\text{P(Black card first)} = \frac{\text{Number of black cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2}\][/tex]
3. Since the card is replaced, determine the probability of drawing a heart second:
- The probability of drawing a heart from the deck is given by the number of hearts divided by the total number of cards.
- [tex]\[\text{P(Heart second)} = \frac{\text{Number of hearts}}{\text{Total number of cards}} = \frac{13}{52} = \frac{1}{4}\][/tex]
4. Calculate the combined probability of both events happening:
- The events are independent because the cards are replaced after each pick. So, the combined probability is the product of the two individual probabilities.
- [tex]\[\text{Combined probability} = \text{P(Black card first)} \times \text{P(Heart second)} = \left(\frac{1}{2}\right) \times \left(\frac{1}{4}\right) = \frac{1}{8}\][/tex]
Therefore, the probability that a black card is chosen first and a heart is chosen second is [tex]\(\frac{1}{8}\)[/tex].
The correct answer is:
[tex]\[\boxed{\frac{1}{8}}\][/tex]
