Question

The interarrival time between customers to a restaurant is Exp(0.1) in minutes (including the time between their opening and arrival of the 1st customer). what is the probability that the first 5 customers arrive within 40 min?

Answer 1

The interarrival time between customers to a restaurant is given to follow Exp(0.1) in minutes, i.e. it takes 10 minutes between arrivals on an average.

Now we want to find the probability of the first 5 arrivals within 40 minutes. i.e. if T denotes the time taken for the first 5 arrivals then T follows a Gamma distribution with shape parameter 5 and rate parameter 0.1.

Here we want to find P(T<40)

Thus using the R command "pgamma(40,5,1/10)" the corresponding probability is 0.3711631.