A complicated mechanical system contains 8 components. If there is a 18% chance that each component will fail during processing, and components fail independently of each other, then (a) What is the probability that fewer than 3 components will fail?
(b) What is the meet likely number of failures?

(a) The probability that fewer than 3 components will fail is  0.8392.

(b) The mean likely number of failures is 1.44.

Step-by-step explanation:

We are given that a complicated mechanical system contains 8 components. If there is an 18% chance that each component will fail during processing, and components fail independently of each other.

Let X = Number of components fail during processing

The above situation can be represented through the binomial distribution;

where, n = number of samples (trials) taken = 8 components

            r = number of success = fewer than 3 components will fail

           p = probability of success which in our question is the probability

                 that each component will fail during processing, i.e. p = 18%

SO, X ~ Binom(n = 8, p = 0.18)

(a) The probability that fewer than 3 components will fail is given by = P(X < 3)

P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)

=

=

= 0.8392

(b) The mean likely number of failures is given by the following formula;

      Mean of X, E(X) = n p

                                 = = 1.44

Hence, the mean likely number of failures is 1.44.