A new phone answering system for a company is capable of han

A new phone answering system for a company is capable of handling 9 calls every 10 mins. Prior to installing the new system, company analysist determine that the incoming calls to the system are poisson distrubuted with a mean equal to 5 every 10 mins. If this incoming call distrubution is what the anaylsist think it is, what is the probability that in a 10 min period more calls will arrive than the system can handle.

Solution

It will overflow if more than 9 calls come in 10 minutes.

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    5      
          
x = our critical value of successes =    9      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   9   ) =    0.968171943
          
Thus, the probability of at least   10   successes is  
          
P(more than   9   ) =    0.031828057 [ANSWER]