Question

An optical inspection system is used to distinguish among different part types. The probability of a correct classification of any part is 0.96. Suppose that three parts are inspected and that the classifications are independent. Let the random variable X denote the number of parts that are correctly classified. Determine the probability mass function of X. Round your answers to four decimal places (e.g. 98.7654).

Answer 1

Probability Mass Function:

x: 0 1 2 3

P(x): 0.000064 0.004608 0.115902 0.884736

Explanation:

We are given the following information:

We treat correct classification as a success.

P(correct classification) = 0.96

Then the number of classification follows a binomial distribution, where

`P(X=x) = \binom{n}{x}.p^x.(1-p)^(n-x)`

where n is the total number of observations, x is the number of success, p is the probability of success.

Now, we are given n = 3 and x = 0, 1, 2, 3

We have to evaluate:

`P(x = 0)\\= \binom{3}{0}(0.96)^0(1-0.96)^3\\=0.000064`

`P(x = 1)\\= \binom{3}{1}(0.96)^1(1-0.96)^2\\=0.004608`

`P(x = 2)\\= \binom{3}{2}(0.96)^2(1-0.96)^1\\=0.115902`

`P(x = 3)\\= \binom{3}{3}(0.96)^3(1-0.96)^0\\=0.884736`

PMF:

x: 0 1 2 3

P(x): 0.000064 0.004608 0.115902 0.884736