Final answer:
To find the probability that a person waits fewer than 18.5 minutes for a bus with a uniform distribution between 0 and 25 minutes, we calculate the area under the probability density function (PDF) curve. By integrating the PDF from 0 to 18.5 minutes, we find a probability of 0.74 or 74%.
Step-by-step explanation:
To find the probability that a person waits fewer than 18.5 minutes, we need to calculate the area under the probability density function (PDF) curve for the given uniform distribution. The PDF for a uniform distribution is constant over the interval of the distribution, and can be represented by the formula:
f(x) = 1 / (b - a)
Where 'a' and 'b' are the lower and upper limits of the distribution. In this case, a = 0 and b = 25. Thus, the PDF becomes:
f(x) = 1 / (25 - 0) = 1 / 25 = 0.04
To find the probability of waiting fewer than 18.5 minutes, we need to find the area under the PDF curve from 0 to 18.5 minutes. This can be done by calculating the integral of the PDF from 0 to 18.5:
P(X < 18.5) = ∫(0 to 18.5) f(x) dx = ∫(0 to 18.5) 0.04 dx
Integrating from 0 to 18.5 gives:
P(X < 18.5) = 0.04 * x |(0 to 18.5) = 0.04 * (18.5 - 0) = 0.04 * 18.5 = 0.74
The probability that a person waits fewer than 18.5 minutes is 0.74 or 74%