Answer :
Let's solve this problem step-by-step.
1. Understand the deck composition:
- A standard deck contains 52 cards.
- There are 26 black cards (13 spades and 13 clubs).
- There are 13 hearts among the red cards.
2. First event: Drawing a black card:
- The number of black cards is 26 out of a total of 52 cards.
- The probability \( P(\text{Black card first}) \) is calculated as follows:
[tex]\[ P(\text{Black card first}) = \frac{\text{Number of black cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2} \][/tex]
3. Second event: Drawing a heart:
- Since the card is replaced, the total number of cards remains 52, and the number of hearts remains 13.
- The probability \( P(\text{Heart second}) \) is calculated as follows:
[tex]\[ P(\text{Heart second}) = \frac{\text{Number of hearts}}{\text{Total number of cards}} = \frac{13}{52} = \frac{1}{4} \][/tex]
4. Combined probability of both events:
- The events are independent because the first card is replaced before drawing the second card.
- The combined probability \( P(\text{Black card first and Heart second}) \) is the product of the individual probabilities:
[tex]\[ P(\text{Black card first and Heart second}) = P(\text{Black card first}) \times P(\text{Heart second}) = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} \][/tex]
Therefore, the probability that the first card chosen is black and the second card is a heart is:
[tex]\[ \boxed{\frac{1}{8}} \][/tex]
1. Understand the deck composition:
- A standard deck contains 52 cards.
- There are 26 black cards (13 spades and 13 clubs).
- There are 13 hearts among the red cards.
2. First event: Drawing a black card:
- The number of black cards is 26 out of a total of 52 cards.
- The probability \( P(\text{Black card first}) \) is calculated as follows:
[tex]\[ P(\text{Black card first}) = \frac{\text{Number of black cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2} \][/tex]
3. Second event: Drawing a heart:
- Since the card is replaced, the total number of cards remains 52, and the number of hearts remains 13.
- The probability \( P(\text{Heart second}) \) is calculated as follows:
[tex]\[ P(\text{Heart second}) = \frac{\text{Number of hearts}}{\text{Total number of cards}} = \frac{13}{52} = \frac{1}{4} \][/tex]
4. Combined probability of both events:
- The events are independent because the first card is replaced before drawing the second card.
- The combined probability \( P(\text{Black card first and Heart second}) \) is the product of the individual probabilities:
[tex]\[ P(\text{Black card first and Heart second}) = P(\text{Black card first}) \times P(\text{Heart second}) = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} \][/tex]
Therefore, the probability that the first card chosen is black and the second card is a heart is:
[tex]\[ \boxed{\frac{1}{8}} \][/tex]
