Question
What is the probability of randomly selecting a card from a standard 52 card deck and having the card be a black card or a face card?
A.
1/2
B.
3/13
C.
8/13
D.
3/26
A.
1/2
B.
3/13
C.
8/13
D.
3/26
Answers
GPT-4o mini
To determine the probability of randomly selecting a card from a standard 52-card deck and having the card be a black card or a face card, we first need to identify how many black cards and face cards are present in the deck.
1. **Total number of cards in a standard deck**: 52
2. **Black cards**: There are two suits that are black (spades and clubs), each containing 13 cards. Therefore, the total number of black cards is:
\[
13 \, (\text{spades}) + 13 \, (\text{clubs}) = 26 \, \text{black cards}
\]
3. **Face cards**: Each suit (hearts, diamonds, clubs, spades) has 3 face cards (Jack, Queen, King). Thus, the total number of face cards is:
\[
3 \, (\text{face cards per suit}) \times 4 \, (\text{suits}) = 12 \, \text{face cards}
\]
4. **Overlap between black cards and face cards**: The black suits (spades and clubs) also contain face cards. Specifically:
- Spades: Jack, Queen, King (3 face cards)
- Clubs: Jack, Queen, King (3 face cards)
Thus, the total number of black face cards is:
\[
3 \, (\text{face cards in spades}) + 3 \, (\text{face cards in clubs}) = 6 \, \text{black face cards}
\]
5. **Using the principle of inclusion-exclusion** to avoid double counting, we calculate the total number of cards that are either black or face cards:
\[
\text{Total black cards} + \text{Total face cards} - \text{Black face cards}
\]
\[
= 26 \, (\text{black cards}) + 12 \, (\text{face cards}) - 6 \, (\text{black face cards}) = 32
\]
6. **Probability**: The probability of selecting a black card or a face card is the ratio of the number of favorable outcomes to the total number of outcomes (total cards in the deck):
\[
P(\text{black card or face card}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{32}{52}
\]
7. **Simplifying the fraction**:
\[
\frac{32}{52} = \frac{8}{13}
\]
Hence, the probability of randomly selecting a card from a standard 52-card deck and having the card be a black card or a face card is:
**C. \( \frac{8}{13} \)**
1. **Total number of cards in a standard deck**: 52
2. **Black cards**: There are two suits that are black (spades and clubs), each containing 13 cards. Therefore, the total number of black cards is:
\[
13 \, (\text{spades}) + 13 \, (\text{clubs}) = 26 \, \text{black cards}
\]
3. **Face cards**: Each suit (hearts, diamonds, clubs, spades) has 3 face cards (Jack, Queen, King). Thus, the total number of face cards is:
\[
3 \, (\text{face cards per suit}) \times 4 \, (\text{suits}) = 12 \, \text{face cards}
\]
4. **Overlap between black cards and face cards**: The black suits (spades and clubs) also contain face cards. Specifically:
- Spades: Jack, Queen, King (3 face cards)
- Clubs: Jack, Queen, King (3 face cards)
Thus, the total number of black face cards is:
\[
3 \, (\text{face cards in spades}) + 3 \, (\text{face cards in clubs}) = 6 \, \text{black face cards}
\]
5. **Using the principle of inclusion-exclusion** to avoid double counting, we calculate the total number of cards that are either black or face cards:
\[
\text{Total black cards} + \text{Total face cards} - \text{Black face cards}
\]
\[
= 26 \, (\text{black cards}) + 12 \, (\text{face cards}) - 6 \, (\text{black face cards}) = 32
\]
6. **Probability**: The probability of selecting a black card or a face card is the ratio of the number of favorable outcomes to the total number of outcomes (total cards in the deck):
\[
P(\text{black card or face card}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{32}{52}
\]
7. **Simplifying the fraction**:
\[
\frac{32}{52} = \frac{8}{13}
\]
Hence, the probability of randomly selecting a card from a standard 52-card deck and having the card be a black card or a face card is:
**C. \( \frac{8}{13} \)**