Refer to Exercise 17 for the definition of a k out of n system. For a certain 4 out of 6 system, assume that on a rainy day each component has probability 0.7 of functioning, and that on a nonrainy day each component has probability 0.9 of functioning.

a. What is the probability that the system functions on a rainy day?

b. What is the probability that the system functions on a nonrainy day?

c. Assume that the probability of rain tomorrow is 0.20. What is the probability that the system will function tomorrow?

Solution 18E

Step1 of 4:

Let us consider a random variable X it presents the number of components function properly on a rainy season.Then X follows binomial distribution with parameters “n and p” that is X B(n, p),

The probability mass function of binomial distribution is given by

, x = 0,1,2,...,n.

Where,

n = sample size

= 6

x = random variable

p = probability of success

= Probability that the each component is function on rainy season

= 0.70

q = 1 - p (probability of failure)

= 1 - 0.70

= 0.30

Here our goal is:

a.We need to find the probability that the system functions on a rainy day.

b.We need to find the probability that the system functions on a non-rainy day.

c.We need to find the probability that the system will function tomorrow, by assuming that the probability of rain tomorrow is 0.20.

Step2 of 4:

a).

We have n = 6 and p = 0.70.

P(The system functions on a rainy day) = P(X 4)

= P(X = 4) + P(X = 5) + P(X = 6)

= {++

}

= 15(0.2401)(0.09)+6(0.1680)(0.30)+1(0.1176)(1)

= 0.3241+0.3024+0.1176

= 0.7441

Hence, P(The system functions on a rainy day) = 0.7441.

Step3 of 4:

b).

We have n = 6 and...