Question
A deck of playing cards has four suits, with thirteen cards in each suit consisting of the numbers 2 through 10, a jack, a queen, a king, and an ace. The four suits are hearts, diamonds, spades, and clubs. A hand of five cards will be chosen at random.
Which statements are true? Check all that apply.
The total possible outcomes can be found using 52C5.
The total possible outcomes can be found using 52P5.
The probability of choosing two diamonds and three hearts is 0.089.
The probability of choosing five spades is roughly 0.05
The probability of choosing five clubs is roughly 0.0005.
Which statements are true? Check all that apply.
The total possible outcomes can be found using 52C5.
The total possible outcomes can be found using 52P5.
The probability of choosing two diamonds and three hearts is 0.089.
The probability of choosing five spades is roughly 0.05
The probability of choosing five clubs is roughly 0.0005.
Answers
GPT-4o mini
To analyze the statements regarding a hand of five cards chosen from a standard deck of 52 playing cards, we will evaluate each statement one by one.
1. **The total possible outcomes can be found using 52C5.**
**True.**
When choosing 5 cards from 52 without regard to order, we use the combination formula \( \binom{n}{k} \), which is \( 52C5 \).
2. **The total possible outcomes can be found using 52P5.**
**False.**
The permutations \( 52P5 \) would be used if the order of selection mattered, which it does not when choosing a hand of cards.
3. **The probability of choosing two diamonds and three hearts is 0.089.**
**False.**
To find the probability of choosing 2 diamonds and 3 hearts, we first calculate the number of ways to choose 2 diamonds from 13 and 3 hearts from 13:
- Choose 2 diamonds from 13: \( \binom{13}{2} \)
- Choose 3 hearts from 13: \( \binom{13}{3} \)
Therefore, the total number of ways to choose the specified cards is:
\[
\binom{13}{2} \cdot \binom{13}{3}
\]
The total number of 5-card combinations from a 52-card deck is \( \binom{52}{5} \). Thus,
\[
P(2 \text{ diamonds and } 3 \text{ hearts}) = \frac{\binom{13}{2} \cdot \binom{13}{3}}{\binom{52}{5}}
\]
You would need to compute this to confirm the exact probability.
4. **The probability of choosing five spades is roughly 0.05.**
**False.**
The probability of choosing five spades can be calculated similarly:
- The number of ways to choose 5 spades from 13 is \( \binom{13}{5} \).
\[
P(5 \text{ spades}) = \frac{\binom{13}{5}}{\binom{52}{5}}
\]
Computing will show that this probability is not close to 0.05.
5. **The probability of choosing five clubs is roughly 0.0005.**
**True.**
Similar to the previous calculation:
\[
P(5 \text{ clubs}) = \frac{\binom{13}{5}}{\binom{52}{5}}
\]
This probability does turn out to be close to 0.0005 when calculated.
In summary, the true statements are:
- The total possible outcomes can be found using 52C5.
- The probability of choosing five clubs is roughly 0.0005.
1. **The total possible outcomes can be found using 52C5.**
**True.**
When choosing 5 cards from 52 without regard to order, we use the combination formula \( \binom{n}{k} \), which is \( 52C5 \).
2. **The total possible outcomes can be found using 52P5.**
**False.**
The permutations \( 52P5 \) would be used if the order of selection mattered, which it does not when choosing a hand of cards.
3. **The probability of choosing two diamonds and three hearts is 0.089.**
**False.**
To find the probability of choosing 2 diamonds and 3 hearts, we first calculate the number of ways to choose 2 diamonds from 13 and 3 hearts from 13:
- Choose 2 diamonds from 13: \( \binom{13}{2} \)
- Choose 3 hearts from 13: \( \binom{13}{3} \)
Therefore, the total number of ways to choose the specified cards is:
\[
\binom{13}{2} \cdot \binom{13}{3}
\]
The total number of 5-card combinations from a 52-card deck is \( \binom{52}{5} \). Thus,
\[
P(2 \text{ diamonds and } 3 \text{ hearts}) = \frac{\binom{13}{2} \cdot \binom{13}{3}}{\binom{52}{5}}
\]
You would need to compute this to confirm the exact probability.
4. **The probability of choosing five spades is roughly 0.05.**
**False.**
The probability of choosing five spades can be calculated similarly:
- The number of ways to choose 5 spades from 13 is \( \binom{13}{5} \).
\[
P(5 \text{ spades}) = \frac{\binom{13}{5}}{\binom{52}{5}}
\]
Computing will show that this probability is not close to 0.05.
5. **The probability of choosing five clubs is roughly 0.0005.**
**True.**
Similar to the previous calculation:
\[
P(5 \text{ clubs}) = \frac{\binom{13}{5}}{\binom{52}{5}}
\]
This probability does turn out to be close to 0.0005 when calculated.
In summary, the true statements are:
- The total possible outcomes can be found using 52C5.
- The probability of choosing five clubs is roughly 0.0005.