Problem #2 3.86 (change it looks like quiz 2) An electronic product contains 40

Question

Problem #2 3.86 (change it looks like quiz 2)

An electronic product contains 40 integrated circuits. The probability that any integrated circuit is defective is 0.01, and the integrated circuits are independent. The product operates only if at least 95% of integrated circuits are not defective.
(a) What is the probability mass function of the number of defective circuits (2pt).
(b) What is the probability that the product operates? (4pt)
(c) If someone buys three products, what is the probability that exactly one of them will not operate? (4pt)

Explanation / Answer

Solution:-

a)

n = 40, p = 0.01

The distribution follows a binomial, so

By applying binomial distribution:-

P(x, n, p) = nCx*px *(1 - p)(n - x)

b) The probability that the product operates is 0.9925

The product operates only if at least 95% of integrated circuits are not defective.

So 95% of the 40 = (95/100)*40 = 38

The probability that a circuit is not defective = 1 - 0.01 = 0.99

p = 0.99, n = 40, x = 38

By applying binomial distribution:-

P(x, n, p) = nCx*px *(1 - p)(n - x)

P(x > 38) = 0.9925

The probability that the product operates is 0.9925

c) The probability that exactly one of them will not operate is P(x = 1) = 0.02216.

The probability that product will not operates = 1 - 0.9925 = 0.0075

p = 0.0075, n = 3, x = 1

By applying binomial distribution:-

P(x, n, p) = nCx*px *(1 - p)(n - x)

P(x = 1) = 0.02216

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