A phone company is looking to build a tower near a rural town to provide service

Question

A phone company is looking to build a tower near a rural town to provide service to the community. The local government will approve the project only if the tower can handle at least 90% of the calls placed at noon. Each antenna placed on the tower can handle 3 simultaneous calls. The expected number of calls simultaneously placed at noon is 2. How many antennae should the cell phone company place on the tower to ensure that all calls placed at noon can be handled with at least 90% probability? (Assume that each call placed in the rural community is independent of any other call placed. Also, assume the size of the rural community is between 20 and 50 inclusive, and the probability that any individual makes a call is less than 0.05. Notice that we do not know the exact number of people living in the community, nor do we know the exact probability of any one of them placing a call at noon.) A discovery of mineral deposits in the area has caused the population of the town to double. The probability that each resident makes a call at noon remains the same. What is the probability that a tower with only one antenna can handle all calls placed at noon?

Explanation / Answer

a)

In this case, we would take =50(0.05)= 2.5

We would be doing it for x=2 as has been mentioned in the question

P(x; ) = (e^- ) (^x) / x!

probability is 0.2565

b)

In this case, we would take =50(0.05)= 2.5

We would be doing it for x<=3 as the population has doubled

P(x; ) = (e^- ) (^x) / x!

probability is 0.7575

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