Allen Associates write weekend trip insurance at a very nom

Allen & Associates write weekend trip insurance at a very nominal charge. Records show that the probability a motorist will have an accident during the weekend and will file a claim is quite small (.0005). Suppose Alden wrote 400 policies for the forthcoming weekend. Compute the probability that exactly two claims will be filed using the equation for Poisson Probability. Note: The symbol X is the mean (expected value) which we used as mu=np. So lambda is nothing more than the mean number of occurrences (successes - np) in a particular interval. Get the probability that the number of claims is at least 3 from Poisson Tables.

Solution

A)

Note that the probability of x successes out of n trials is          
          
P(x) = u^x e^(-u) / x!          
          
where          
          
u = the mean number of successes = np = 400*0.0005 =   0.2      
          
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) =    0.016374615 [ANSWER]

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B)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    0.2      
          
x = our critical value of successes =    3      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   2   ) =    0.998851519
          
Thus, the probability of at least   3   successes is  
          
P(at least   3   ) =    0.001148481 [ANSWER]