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)

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.

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 \( b - a = 8 - 0 = 8 \).

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 \( b - \text{threshold} = 8 - 1.25 = 6.75 \).

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:

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

Round the probability to three decimal places:

\[ 0.844 \]

Final Answer