Answer :
To find the probability that the first card chosen is a black card and the second card chosen is a heart, we need to follow these steps:
1. Determine the total number of cards in the deck:
A standard deck has 52 cards.
2. Calculate the probability of drawing a black card first:
- There are 26 black cards in a deck (comprising of spades and clubs).
- The probability of drawing a black card first is given by the ratio of black cards to the total number of cards.
[tex]\[ \text{Probability of drawing a black card first} = \frac{26}{52} = 0.5 \][/tex]
3. Calculate the probability of drawing a heart second:
- There are 13 hearts in a deck.
- Since the cards are replaced after each pick, the total number of cards remains 52 after the first card is drawn.
- The probability of drawing a heart second is given by the ratio of hearts to the total number of cards.
[tex]\[ \text{Probability of drawing a heart second} = \frac{13}{52} = 0.25 \][/tex]
4. Determine the combined probability of both events happening sequentially:
- The events are independent because the card is replaced after each draw.
- To find the combined probability of both events occurring (drawing a black card first and a heart second), we multiply the probabilities of each event.
[tex]\[ \text{Combined probability} = \left(\frac{26}{52}\right) \times \left(\frac{13}{52}\right) = 0.5 \times 0.25 = 0.125 \][/tex]
The combined probability can also be converted to a fraction:
[tex]\[ 0.125 = \frac{1}{8} \][/tex]
Therefore, the probability that a black card is chosen first and a heart is chosen second is \(\frac{1}{8}\), which matches the given choices. Thus, the correct answer is:
[tex]\[ \boxed{\frac{1}{8}} \][/tex]
1. Determine the total number of cards in the deck:
A standard deck has 52 cards.
2. Calculate the probability of drawing a black card first:
- There are 26 black cards in a deck (comprising of spades and clubs).
- The probability of drawing a black card first is given by the ratio of black cards to the total number of cards.
[tex]\[ \text{Probability of drawing a black card first} = \frac{26}{52} = 0.5 \][/tex]
3. Calculate the probability of drawing a heart second:
- There are 13 hearts in a deck.
- Since the cards are replaced after each pick, the total number of cards remains 52 after the first card is drawn.
- The probability of drawing a heart second is given by the ratio of hearts to the total number of cards.
[tex]\[ \text{Probability of drawing a heart second} = \frac{13}{52} = 0.25 \][/tex]
4. Determine the combined probability of both events happening sequentially:
- The events are independent because the card is replaced after each draw.
- To find the combined probability of both events occurring (drawing a black card first and a heart second), we multiply the probabilities of each event.
[tex]\[ \text{Combined probability} = \left(\frac{26}{52}\right) \times \left(\frac{13}{52}\right) = 0.5 \times 0.25 = 0.125 \][/tex]
The combined probability can also be converted to a fraction:
[tex]\[ 0.125 = \frac{1}{8} \][/tex]
Therefore, the probability that a black card is chosen first and a heart is chosen second is \(\frac{1}{8}\), which matches the given choices. Thus, the correct answer is:
[tex]\[ \boxed{\frac{1}{8}} \][/tex]
