I am not sure how to approach this problem. For the first part how do I deal with time? Do I consider the probability of no buses arriving within an hour? Do I consider lambda to be 1/15 here?
A bus route in a large town has one bus scheduled every 15 minutes. Traffic conditions in the town are such that the arrival times of buses at a particular bus stop may be assumed to follow a Poisson process. Mr. James arrives at the bus stop at 12 midday to find no bus at the stop. He intends to get on the first bus to arrive.
1) What is the probability that the first bus will not have arrived by 1:00 pm the same day?
The first bus arrived at 1:10 pm but was full, so Mr. James was unable to board it.
ii. What is the probability that at least two more buses will arrive between 1:10 pm and 1:20 pm.
As already noted in a period of 1 hour your distribution is $Po(4)$ thus the probability that the first bus arrive not early then 1 pm is the probability that 0 buses arrive in the period so
$$P(X=0)=e^{-4}$$
Can you proceed by yourself for the 2nd question?
EDIT: your answer for the 1st question is correct. For the 2nd one it's correct too...but there's an obvious typo. The correct one is
$$1-\frac{5}{3}e^{-\frac{2}{3}}$$