Problem 16E
Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.
Precipitation In Savannah, Georgia, the mean number of days in July with 0.01 inch or more of precipitation is 13. Find the probability that, next July, the number of days with 0.01 inch or more of precipitation in Savannah is (a) exactly 16 days, (b) at most 16 days, and (c) more than 16 days.
Solution 16E
Step1 of 4:
From the given problem we have an experiment, In that the mean number of days in July with 0.01 inch or more of precipitation is 13.
Here our goal is:
a). We need to find the probability that the number of days with 0.01 inch or more of precipitation in Savannah is exactly 16 days.
b). We need to find the probability that the number of days with 0.01 inch or more of precipitation in Savannah is at most 16 days.
c). We need to find the probability that the number of days with 0.01 inch or more of precipitation in Savannah is more than 16 days.
Step2 of 4:
a).
Let,
Let us consider a random variable ‘x’ it presents the number of days in July with 0.01 inch or more of precipitation. And here ‘x’ follows the Poisson distribution with parameters ‘’. The probability mass function of the Poisson distribution is:
The probability that the number of days with 0.01 inch or more of precipitation in Savannah is exactly 16 days is:
Therefore, P(X = 16) = 0.072.
Step3 of 4:
b).
The probability that the probability that the number of days with 0.01 inch or more of precipitation in Savannah is at most 16 days is:
Where,
is obtained from excel by using the function:
“=Poisson(x, false)”.
X 

0 
2.26033E06 
1 
2.93843E05 
2 
0.000190998 
3 
0.000827657 
4 
0.002689886 
5 
0.006993704 
6 
0.015153025 
7 
0.028141333 
8 
0.045729666 
9 
0.066053962 
10 
0.085870151 
11 
0.101482905 
12 
0.109939814 
13 
0.109939814 
14 
0.10208697 
15 
0.088475374 
16 
0.071886242 
Total 
0.835493148 
Now,
Therefore,
Step4 of 4:
c).
The probability that the probability that the number of days with 0.01 inch or more of precipitation in Savannah is more than 16 days is:
From part (b), we have
Therefore,