You arrive at a bus stop at 10 o'clock, knowing that the bus...

You arrive at a bus stop at 10 o'clock, knowing that the bus will arrive at some time uniformly distributed between 10 and $10: 30$. (a) What is the probability that you will have to wait longer than 10 minutes? (b) If at 10:15 the bus has not yet arrived, what is the probability that you will have to wait at least an additional 10 minutes?

You arrive at a bus stop at 10 o'clock, knowing that the bus will arrive at some time uniformly distributed between 10 and $10: 30$.
(a) What is the probability that you will have to wait longer than 10 minutes?
(b) If at 10:15 the bus has not yet arrived, what is the probability that you will have to wait at least an additional 10 minutes?

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Key Concepts

Uniform Distribution

This is a probability distribution where all outcomes in a given interval are equally likely. The probability of the random variable falling within any sub-interval is proportional to the length of that sub-interval relative to the total interval. It is often used in scenarios where there is no reason to favor one outcome over another within a range.

Conditional Probability

This concept involves computing the probability of an event given that another event has already occurred. It adjusts the original sample space to include only the outcomes consistent with the given information, thereby affecting the computation of likelihoods for subsequent events.

Waiting Time Problems

These problems analyze the time a person waits for an event to occur, typically modeled as a continuous random variable. They often utilize distributions, such as the uniform distribution, to evaluate probabilities related to delays or waiting periods, especially when the timing of events is assumed to be random over a specified interval.

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