Final answer:
To find the probability of more than 3 people arriving at checkout during 2 minutes with a mean arrival rate of 0.5 per minute, we use the Poisson distribution to calculate the probability of 0, 1, 2, and 3 arrivals, then subtract the total from 1. For time between arrivals, the exponential distribution is used.
Step-by-step explanation:
The student's question deals with finding the probability of a certain number of people arriving at a store checkout during a given period, assuming arrivals follow a Poisson process. Since people are arriving independently and at a constant mean rate of 0.5 per minute, we can determine the probability of more than 3 people arriving in 2 minutes using the formula for the Poisson distribution:
P(X > k) = 1 - (P(X=0) + P(X=1) + P(X=2) + P(X=3))
where P(X=k) is the probability of exactly k events occurring in a fixed interval of time. Given the mean rate (\(\lambda\)) of 0.5 per minute, for 22 minutes mean rate would be \(\lambda\times t\) where t is 2 minutes, so \(\lambda = 1\). Here, we calculate the probabilities for 0, 1, 2, and 3 people arriving using the Poisson probability mass function and then subtract the sum of these probabilities from 1 to find the probability of getting more than 3 arrivals.
In an exponential distribution, the amount of time between two successive events (like customer arrivals) is also of interest. The question implies the use of an exponential distribution to calculate the waiting times between customer arrivals. For instance, if we know customers arrive on average every two minutes (0.5 customers per minute), the probability that the time between two successive arrivals is less than one minute can be found using the exponential distribution formula.