Questions: The waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 8 minutes. Find the probability that a randomly selected passenger has a waiting time greater than 1.25 minutes. Find the probability that a randomly selected passenger has a waiting time greater than 1.25 minutes. (Simplify your answer. Round to three decimal places as needed)

The waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 8 minutes. Find the probability that a randomly selected passenger has a waiting time greater than 1.25 minutes. Find the probability that a randomly selected passenger has a waiting time greater than 1.25 minutes. (Simplify your answer. Round to three decimal places as needed)
Transcript text: The waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 8 minutes. Find the probability that a randomly selected passenger has a waiting time greater than 1.25 minutes. Find the probability that a randomly selected passenger has a waiting time greater than 1.25 minutes. (Simplify your answer. Round to three decimal places as needed)
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Solution

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Solution Steps

To solve this problem, we need to understand that the waiting times are uniformly distributed between 0 and 8 minutes. This means that the probability density function is constant over this interval. To find the probability that a randomly selected passenger has a waiting time greater than 1.25 minutes, we calculate the area under the uniform distribution curve from 1.25 to 8 minutes and divide it by the total area from 0 to 8 minutes.

Step 1: Understand the Uniform Distribution

The waiting times are uniformly distributed between 0 and 8 minutes. This means the probability density function is constant over this interval. The total length of the interval is \( b - a = 8 - 0 = 8 \).

Step 2: Calculate the Probability

To find the probability that a randomly selected passenger has a waiting time greater than 1.25 minutes, we calculate the length of the interval from 1.25 to 8 minutes, which is \( b - \text{threshold} = 8 - 1.25 = 6.75 \).

Step 3: Compute the Probability

The probability is given by the ratio of the length of the interval where the waiting time is greater than 1.25 minutes to the total length of the interval. Thus, the probability is:

\[ \frac{b - \text{threshold}}{b - a} = \frac{6.75}{8} = 0.84375 \]

Step 4: Round the Probability

Round the probability to three decimal places:

\[ 0.844 \]

Final Answer

The probability that a randomly selected passenger has a waiting time greater than 1.25 minutes is \(\boxed{0.844}\).

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