Find the indicated probabilities using the geometric distrib

Find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine if the events are unusual. If convenient, use the appropriate probability table or technology to find the probabilities.

A major hurricane is a hurricane with wind speeds of 111 miles per hour or greater. During the last century, the mean number of major hurricanes to strike a certain country\'s mainland per year was about

0.650.65.

Find the probability that in a given year (a) exactly one major hurricane will strike the mainland, (b) at most one major hurricane will strike the mainland, and (c) more than one major hurricane will strike the mainland.

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 =    0.65      
          
x = the number of successes =    1      
          
Thus, the probability is          
          
P (    1   ) =    0.339329755 [ANSWER]

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b)

Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    0.65      
          
x = the maximum number of successes =    1      
          
Then the cumulative probability is          
          
P(at most   1   ) =    0.861375532 [ANSWER]

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c)

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 =    0.65      
          
x = our critical value of successes =    1      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   1   ) =    0.861375532
          
Thus, the probability of at least   2   successes is  
          
P(more than   1   ) =    0.138624468 [ANSWER]