Question
An experiment involves rolling a die and drawing a card from a standard deck of 52 cards.
What is the probability that the die lands on 4 and the card is a heart?
Answers
GPT-4o mini
To find the probability that the die lands on 4 and the card drawn from a standard deck of cards is a heart, we first determine the probabilities of each individual event.
1. **Probability of rolling a 4 on a die:**
A standard die has 6 faces, each representing a different number (1 through 6). The probability of rolling a specific number (in this case, 4) is:
\[
P(\text{rolling a 4}) = \frac{1}{6}
\]
2. **Probability of drawing a heart from a deck of cards:**
A standard deck contains 52 cards, of which 13 are hearts. The probability of drawing a heart is:
\[
P(\text{drawing a heart}) = \frac{13}{52} = \frac{1}{4}
\]
3. **Calculating the combined probability:**
Since the dice roll and the card draw are independent events, we can multiply their probabilities to find the combined probability:
\[
P(\text{rolling a 4 and drawing a heart}) = P(\text{rolling a 4}) \times P(\text{drawing a heart}) = \frac{1}{6} \times \frac{1}{4}
\]
Calculating the product:
\[
P(\text{rolling a 4 and drawing a heart}) = \frac{1}{6} \times \frac{1}{4} = \frac{1}{24}
\]
Thus, the probability that the die lands on 4 and the card drawn is a heart is \(\frac{1}{24}\).
1. **Probability of rolling a 4 on a die:**
A standard die has 6 faces, each representing a different number (1 through 6). The probability of rolling a specific number (in this case, 4) is:
\[
P(\text{rolling a 4}) = \frac{1}{6}
\]
2. **Probability of drawing a heart from a deck of cards:**
A standard deck contains 52 cards, of which 13 are hearts. The probability of drawing a heart is:
\[
P(\text{drawing a heart}) = \frac{13}{52} = \frac{1}{4}
\]
3. **Calculating the combined probability:**
Since the dice roll and the card draw are independent events, we can multiply their probabilities to find the combined probability:
\[
P(\text{rolling a 4 and drawing a heart}) = P(\text{rolling a 4}) \times P(\text{drawing a heart}) = \frac{1}{6} \times \frac{1}{4}
\]
Calculating the product:
\[
P(\text{rolling a 4 and drawing a heart}) = \frac{1}{6} \times \frac{1}{4} = \frac{1}{24}
\]
Thus, the probability that the die lands on 4 and the card drawn is a heart is \(\frac{1}{24}\).