QuestionJune 23, 2025

The arrival time of an elevator in a 12-story dormitory is equally likely at any time during the next 4 minutes. a. Calculate the expected arrival time. Note: Round your answer to 2 decimal places. Expected arrival time square b. What is the probability that an elevator arrives in less than 11/2 minutes? Note: Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places. Probability square c. What is the probability that the wait for an elevator is more than 11/2 minutes? Note: Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places. Probability square

The arrival time of an elevator in a 12-story dormitory is equally likely at any time during the next 4 minutes. a. Calculate the expected arrival time. Note: Round your answer to 2 decimal places. Expected arrival time square b. What is the probability that an elevator arrives in less than 11/2 minutes? Note: Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places. Probability square c. What is the probability that the wait for an elevator is more than 11/2 minutes? Note: Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places. Probability square
The arrival time of an elevator in a 12-story dormitory is equally likely at any time during the next 4 minutes. a. Calculate the expected arrival time. Note: Round your answer to 2 decimal places. Expected arrival time square b. What is the probability that an elevator arrives in less than 11/2 minutes? Note: Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places. Probability square c. What is the probability that the wait for an elevator is more than 11/2 minutes? Note: Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places. Probability square

Solution
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Answer

Expected arrival time: 2.00 ### Probability (less than 11/2 minutes): 1.000 ### Probability (more than 11/2 minutes): 0.000 Explanation 1. Calculate the expected arrival time For a uniform distribution over [0, 4], the expected value is the midpoint. **Expected Value = \frac{a + b}{2}** where a = 0 and b = 4. So, \frac{0 + 4}{2} = 2. 2. Calculate probability for arrival in less than 11/2 minutes Convert 11/2 to decimal: 11/2 = 5.5. Probability is area under the curve from 0 to 5.5. Since 5.5 > 4, it covers the entire interval. **Probability = \frac{\text{Length of Interval}}{\text{Total Length}} = \frac{4}{4} = 1**. 3. Calculate probability for wait more than 11/2 minutes Since 11/2 = 5.5 exceeds the interval [0, 4], probability is zero. **Probability = \frac{\text{Length of Interval beyond 4}}{\text{Total Length}} = \frac{0}{4} = 0**.

Explanation

1. Calculate the expected arrival time<br /> For a uniform distribution over $[0, 4]$, the expected value is the midpoint. **Expected Value = \frac{a + b}{2}** where $a = 0$ and $b = 4$. So, $\frac{0 + 4}{2} = 2$.<br />2. Calculate probability for arrival in less than $11/2$ minutes<br /> Convert $11/2$ to decimal: $11/2 = 5.5$. Probability is area under the curve from $0$ to $5.5$. Since $5.5 > 4$, it covers the entire interval. **Probability = \frac{\text{Length of Interval}}{\text{Total Length}} = \frac{4}{4} = 1**.<br />3. Calculate probability for wait more than $11/2$ minutes<br /> Since $11/2 = 5.5$ exceeds the interval $[0, 4]$, probability is zero. **Probability = \frac{\text{Length of Interval beyond 4}}{\text{Total Length}} = \frac{0}{4} = 0**.
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