Question

The annual rainfall in Cleveland, Ohio, is approximately a normal random variable with mean 40.2 inches and standard deviation 8.4 inches. What is the probability that (a) next year’s rainfall will exceed 44 inches? (b) the yearly rainfalls in exactly 3 of the next 7 years will exceed 44 inches? Assume that if Ai is the event that the rainfall exceeds 44 inches in year i (from now), then the events Ai, i Ú 1, are independent.

Short answer

The result is

(a) 0.32636

(b) 0.25

Step-by-step solution

Step:1 Given Information

With a mean of 40.2 inches and a standard deviation of 8.4 inches, yearly rainfall in Cleveland, Ohio, is roughly a normal random variable. What is the likelihood that (a) rainfall will reach 44 inches next year? (b) in exactly three of the next seven years, annual rainfall will exceed 44 inches Assume that if Ai is the occurrence that the rainfall in year I (from now) exceeds 44 inches, then Ai, I 1 are independent events.

Step:2 Explanation of the solution

Define the random variable X, which represents Cleveland's annual rainfall. That is given to us

(a)
We are required to find

The probability that each of the next seven years will have more than 44 inches of rainfall is p=0.32636, and it will happen every year independently of the others. As a result, the number of years in which rainfall surpasses 44 inches can be calculated as. As a result, the necessary probability is

$P(Y=3)=\left(\begin{array}{l}7\\ 3\end{array}\right)0.{32636}^3(1-0.32636{)}^4=0.25$