# Probability of two buses being in a bus stop at the same time

Question statement:

The bus type of A arrives to a certain bus stop every 30 minutes and waits for 5 minutes. The bus type of B arrives every 60 minutes and waits for 4 minutes. Probabilities of arrival of these two bus types have uniform density, and they are independent of each other.

A man goes to the bus stop at a random time, and starts waiting. What is the probability that two buses be in the bus stops at the same time (i.e.; their waiting time periods intersect with each other)?

This was a question I came across last year when I was studying probability theory. I couldn't solve it then and I am still not able to solve it. I can't even find a starting point. How are the probability questions of this type solved?

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Let $t$ be the instant that bus A arrives. Then Bus B must arrive in the interval $t-4, t+5$ if you want both bus at the same time. Bus B arrival is uniformly distributed on on interval of amplitude 64 minutes, hence you have to choose 9 minutes out out 64, which is probability $9/64$.