Final answer:
To calculate the probability that a person waits fewer than 12.5 minutes for the bus, we use the uniform distribution's properties, where the probability is found by dividing the waiting time (12.5 minutes) by the total range of times (15 minutes).
Step-by-step explanation:
Calculating the Probability of Waiting Time
The probability problem provided can be solved using the concepts of uniform distribution. The time that a person waits for a bus is uniformly distributed between 0 and 15 minutes. So, the bus arrival time is a uniform random variable, let's denote it as X, where X ~ U(0, 15). The probability density function (pdf) for a uniform distribution is given by f(x) = 1 / (b - a) for a ≤ x ≤ b, in this case, f(x) = 1/15 as b = 15 and a = 0.
To find the probability that a person waits fewer than 12.5 minutes, we would calculate P(X < 12.5) which is the area under the pdf from 0 to 12.5. Since the uniform distribution is a rectangle, the area equals the base times the height. Here the base would be 12.5 minutes and the height is the constant pdf value which is 1/15. Therefore, the probability is P(X < 12.5) = 12.5 / 15.