Question

An inspector working for a manufacturing company has a 99% chance of correctly identifying defective items and a 0.5% chance of incorrectly classifying a good item as defective. The company has evidence that its line produces 1.0% of defective items. Round your answers to five decimal places (e.g. 98.76543). (a) What is the probability that an item selected for inspection is classified as defective?

Answer 1

The probability that an item selected for inspection is classified as defective is 0.01485.

Explanation:

Denote the events as follows:

A₁ = an item produced by the company is defective.

A₂ = an item produced by the company is not defective.

X = an item is classified as defective by the inspector.

Given:

P (X|A₁) = 0.99

P (X|A₂) = 0.005

P (A₁) = 0.01

P (A₂) = 1 - P (A₁) = 1 - 0.01 = 0.99

The law of total probability states that:

`P(B)=P(B|A)P(A)+P(B|C)P(C)`

Use this law to compute the probability that an item selected for inspection is classified as defective as follows:

`P(X)=P(X|A_(1))P(A_(1))+P(X|A_(2))P(A_(2))\\=(0.99*0.01)+(0.005*0.99))\\=0.01485`

Thus, the probability that an item selected for inspection is classified as defective is 0.01485.