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The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. What is the probability that a person waits fewer than 12.5 minutes?

User Lux
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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.

User Zajer
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