Question

An assembly consists of two mechanical components. Suppose that the probabilities that the first and second components meet specifications are 0.87 and 0.84. Assume that the components are independent. Determine the probability mass function of the number of components in the assembly that meet specifications. X

Answer 1

Answer:

P(X = 0) = 0.0208, P(X = 1) = 0.2484, P(X = 2) = 0.7308

Step-by-step explanation:

Let's define the following events

A: the first component meet specification

B: the second component meet specification

P(A) = 0.87

P(B) = 0.84

Let X be the random variable that represents the number of components in the assembly that meet specifications. Because there are only two mechanical components in the assembly, X can only take the values 0, 1, 2.

P(X = 0) = P(the first component does not meet specification and the second component does not meet specification) =  

$P(A^{c}\cap B^{c}) = P(A^{c})P(B^{c})$ (because of independence)  

= (0.13)(0.16) = 0.0208

P(X = 1) = P(only one component meet specification) = P[(the first component meet specification and the second component does not meet specification) or (the first component does not meet specification and the second component meet specification)] =  

$P[(A\cap B^{c})\cup (A^{c}\cap B)] = P(A\cap B^{c}) + P(A^{c}\cap B)=$ (because sets are mutually exclusive)

$P(A)P(B^{c}) + P(A^{c})P(B)=$ (because of independence)

= (0.87)(0.16) + (0.13)(0.84) = 0.2484

P(X = 2) = P(both components meet specifications) =  

$P(A\cap B) = P(A)P(B)$ (because of independence)

= (0.87)(0.84)

= 0.7308