A deck of playing cards has four suits, with thirteen cards Which statements are true? Check all that apply.
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.
The total possible outcomes can be found using
.
The total possible outcomes can be found using
.
The probability of choosing two diamonds and three
hearts is 0.089 .
The probability of choosing five spades is roughly
0.05

Ask by Washington Burgess. in the United States

Jan 22,2025

Question

A deck of playing cards has four suits, with thirteen cards Which statements are true? Check all that apply.
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.
The total possible outcomes can be found using
.
The total possible outcomes can be found using
.
The probability of choosing two diamonds and three
hearts is 0.089 .
The probability of choosing five spades is roughly
0.05

Ask by Washington Burgess. in the United States

Jan 22,2025

UpStudy AI · Accepted Answer

Let's evaluate each statement one by one based on the standard 52-card deck and probability principles.

1. **The total possible outcomes can be found using $$ \binom{52}{5} $$.**

- **True.**
   - **Explanation:** When choosing a hand of five cards where the order does not matter, the number of possible combinations is given by the combination formula $$ \binom{52}{5} $$.

2. **The total possible outcomes can be found using $$ P_{52,5} $$.**

- **False.**
   - **Explanation:** $$ P_{52,5} $$ represents permutations, which count ordered arrangements. Since the order of cards in a hand doesn't matter, combinations are the appropriate method, not permutations.

3. **The probability of choosing two diamonds and three hearts is 0.089.**

- **False.**
   - **Explanation:** 
     - **Calculation:**
       $$
       \text{Probability} = \frac{\binom{13}{2} \times \binom{13}{3}}{\binom{52}{5}} = \frac{78 \times 286}{2,598,960} \approx 0.00858 \text{ or } 0.858\%
       $$
     - The statement claims a probability of 8.9% (0.089), which is significantly higher than the actual probability of approximately 0.858%.

4. **The probability of choosing five spades is roughly 0.05.**

- **False.**
   - **Explanation:**
     - **Calculation:**
       $$
       \text{Probability} = \frac{\binom{13}{5}}{\binom{52}{5}} = \frac{1,287}{2,598,960} \approx 0.000495 \text{ or } 0.0495\%
       $$
     - If "0.05" is interpreted as 5%, it's incorrect. The actual probability is approximately 0.05%, which might be what was intended, but the wording is unclear. Given the context and typical interpretation, this statement is considered **false**.

**Summary of True Statements:**
- **Only the first statement** regarding the total possible outcomes using combinations $$ \binom{52}{5} $$ is true.

Answer: **All statements are false except the first one, which correctly uses combinations to count outcomes.**
Only the first statement is true. The total possible outcomes are calculated using $$ \binom{52}{5} $$.

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