Using the Poisson approximation to the binomial what is the

Using the Poisson approximation to the binomial, what is the probability of no leaks? (Round your answer to 4 decimal places.)

Using the Poisson approximation to the binomial, what is the probability of three or more leaks?(Round your answer to 4 decimal places.)

Solution

There is an average of 1 leak per 1000 meters. Thus, in 2500 meters, we expect 2.5 leaks.

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 =    2.5      
          
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.082084999 [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 =    2.5      
          
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   ) =    0.543813116
          
Thus, the probability of at least   3   successes is  
          
P(at least   3   ) =    0.456186884 [answer]

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c)

again, the expected number of leaks is

E(x) = 2.5 [answer]