I am struggling with two problems using Poisson processes. I would appreciate your help.
(1) The traffic on a one way street shown below may be satisfactorily described by a Poisson process with an average rate of arrival of 10 cars per minute. A driver is on a side street and is waiting to cross this main street. He will cross as soon as he finds a gap of 15 seconds.
(a) Determine the probability that a gap will be longer than 15 seconds. (I think I did this part and I got 0.0005 - is that right?).
(b) What is the probability that the driver will cross at the fourth gap?
(c) Determine the mean number of gaps he has to wait until crossing the main road.
(d) What is the probability that he will cross within the first 4 gaps?
(2) Strong earthquakes occur according to a Poisson process in a city with a mean rate of once per 50 years. There are 3 bridges in the city. When a strong earthquake occurs, there is a probability of 0.3 that a given bridge will collapse. Assume the events of collapse between bridges during a strong earthquake is statistically independent. Also, the events of bridge collapse between earthquakes are also statistically independent. What is the probability of "no bridge collapse from strong earthquakes" during the next 20 years?
Thank you very much in advance!