Question

Suppose a bus arrives at a bus stop every 26 minutes. If you arrive at the bus stop at a random time, what is the probability that you will have to wait at least 4 minutes for the bus?

Answer 1

The time between buses arriving at the stop follows an exponential distribution with a mean of 26 minutes.

To find the probability of waiting at least 4 minutes for the bus, we can use the cumulative distribution function (CDF) of the exponential distribution:

P(waiting at least 4 minutes) = 1 - P(waiting less than 4 minutes)

The probability of waiting less than 4 minutes can be calculated using the CDF:

P(waiting less than 4 minutes) = 1 - e^(-4/26) ≈ 0.146

Therefore, the probability of waiting at least 4 minutes for the bus is:

P(waiting at least 4 minutes) = 1 - 0.146 ≈ 0.854

So the probability of having to wait at least 4 minutes for the bus is about 85.4%.