Final answer:
The probability of a person waiting fewer than 12.5 minutes for a bus, with the arrival times uniformly distributed between zero and 15 minutes, is 5/6 or approximately 83.33%.
Step-by-step explanation:
Finding the Probability of Waiting Time for a Bus
The scenario given pertains to a uniform distribution of bus arrival times, where the maximum time between buses is 15 minutes. To find the probability that a person waits fewer than 12.5 minutes, we consider the total possible wait time and calculate the proportion of this time that is less than 12.5 minutes.
Since the wait time for the bus is uniformly distributed between zero and 15 minutes, the probability density function (pdf) for this uniform distribution is f(x) = 1/15 for 0 ≤ x ≤ 15. To find the probability of waiting less than 12.5 minutes, we integrate the pdf from 0 to 12.5:
P(X < 12.5) = ∫_{0}^{12.5} f(x) dx = ∫_{0}^{12.5} (1/15) dx
When we complete this integration, we get:
P(X < 12.5) = (12.5 - 0) / 15 = 12.5 / 15
Thus, the probability that a person waits fewer than 12.5 minutes for the bus is 12.5/15, which simplifies to 5/6 or approximately 0.8333, meaning there's about an 83.33% chance of waiting less than 12.5 minutes.