A manager of an apartment store reports that the time of a customer on the second floor must wait for the elevator has a uniform distribution ranging from 0 to 4 minutes. If it takes the elevator 15 seconds to go from floor to floor, find the probability that a hurried customer can reach the first floor in less than 3 minutes after pushing the elevator button on the second floor.

A manager of an apartment store reports that the time of a customer on the second floor must wait for the elevator has a uniform distribution ranging from 0 to 4 minutes. The elevator 15 seconds to go from floor to floor,

Let x denotes the waiting time.

It is given that waiting time is uniformly distributed from 0 to 4.

It is given that it takes 15 seconds to go from floor to floor.

Convert 15 seconds into minutes: : [tex]\frac{15}{60}=0.25[/tex] min

Time to reach first floor is uniformly distributed:

[tex]U(0+0.25, 4+0.25)=U(0.25, 4.25)[/tex]

We need to determine the probability that a hurried customer can reach the first floor in less than 3 minutes after pushing the elevator button on the second floor."

So we need to find P(Y < 3)

[tex]P(Y < 3) = \frac{(3 - 0.25)}{(4.25 - 0.25)} =0.6875[/tex]

Hence, the required probability is 0.6875