Question

An optical inspection system is used to distinguish among different part types. The probability of a correct classification of any part is 0.98. 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.

Answer 1

Answer:

P(X=0)=0.000008

P(X=1)=0.00176

P(X=2)=0.057624

P(X=3)=0.941192

Step-by-step explanation:

Probability of correct classification = p = 0.98

Probability of incorrect classification = q = 1 - p = 0.02

The probability of success and failure is the same for all the trials. The trials are independent of each other and the number of trials is fixed i.e. n = 3.

This satisfies all the conditions of a Binomial Experiment. So we can use Binomial experiment to model the probability mass function.

The general formula of a binomial probability is:

$P(X=x)=^{n}C_{x}(p)^{x}q^{n-x}$

Here x denote the number of successes, which can be {0, 1, 2, 3}. So we need to evaluate the above equation for each value of x to determine the probability Mass function of X, as shown below:

$P(X=0)=^{3}C_{0}(0.98)^{0}(0.02)^{3-0}=0.000008\\\\ P(X=1)=^{3}C_{1}(0.98)^{1}(0.02)^{3-1}=0.001176\\\\ P(X=2)=^{3}C_{2}(0.98)^{2}(0.02)^{3-2}=0.057624\\\\ P(X=3)=^{3}C_{3}(0.98)^{3}(0.02)^{3-3}=0.941192$