Question
Find the probability of the given event. Show all work, and your final answer may be an exact value, a decimal to 3 places, or a percentage to the nearest tenth.
A waiter takes lunch orders for 4 people. He forgets which person ordered which meal, and all 4 people ordered something different. If the waiter randomly chooses a person to give each meal, what is the probability that he will serve the correct meal to each perso
A waiter takes lunch orders for 4 people. He forgets which person ordered which meal, and all 4 people ordered something different. If the waiter randomly chooses a person to give each meal, what is the probability that he will serve the correct meal to each perso
Answers
GPT-4o mini
To find the probability that the waiter serves the correct meal to each of the 4 people, we first need to understand the total number of ways he can serve the meals and the number of ways he can serve them correctly.
1. **Total Possible Arrangements**:
Since there are 4 meals and each person ordered a different meal, the total number of ways to serve these meals to the 4 people can be calculated using the factorial of the number of meals, which is \( 4! \).
\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]
2. **Number of Correct Arrangements**:
There is only 1 correct way to serve the meals, where each person receives their correct meal.
3. **Probability Calculation**:
The probability \( P \) that the waiter serves the correct meal to each person is the ratio of the number of correct arrangements to the total possible arrangements.
\[
P = \frac{\text{Number of Correct Arrangements}}{\text{Total Possible Arrangements}} = \frac{1}{24}
\]
4. **Convert to Decimal and Percentage**:
We can express this probability as a decimal or a percentage:
- As a decimal:
\[
\frac{1}{24} \approx 0.04167 \quad \text{(rounded to 5 decimal places)}
\]
- To express it as a percentage, multiply by 100:
\[
0.04167 \times 100 \approx 4.17\%
\]
Rounding this to the nearest tenth of a percent gives \( 4.2\% \).
Thus, the probability that the waiter serves the correct meal to each person is:
\[
\frac{1}{24} \text{ or } 0.042 \text{ (to 3 decimal places) } \text{ or } 4.2\% \text{ (to the nearest tenth)}.
\]
1. **Total Possible Arrangements**:
Since there are 4 meals and each person ordered a different meal, the total number of ways to serve these meals to the 4 people can be calculated using the factorial of the number of meals, which is \( 4! \).
\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]
2. **Number of Correct Arrangements**:
There is only 1 correct way to serve the meals, where each person receives their correct meal.
3. **Probability Calculation**:
The probability \( P \) that the waiter serves the correct meal to each person is the ratio of the number of correct arrangements to the total possible arrangements.
\[
P = \frac{\text{Number of Correct Arrangements}}{\text{Total Possible Arrangements}} = \frac{1}{24}
\]
4. **Convert to Decimal and Percentage**:
We can express this probability as a decimal or a percentage:
- As a decimal:
\[
\frac{1}{24} \approx 0.04167 \quad \text{(rounded to 5 decimal places)}
\]
- To express it as a percentage, multiply by 100:
\[
0.04167 \times 100 \approx 4.17\%
\]
Rounding this to the nearest tenth of a percent gives \( 4.2\% \).
Thus, the probability that the waiter serves the correct meal to each person is:
\[
\frac{1}{24} \text{ or } 0.042 \text{ (to 3 decimal places) } \text{ or } 4.2\% \text{ (to the nearest tenth)}.
\]