Question

An assembly consists of two mechanical components. Suppose that the probabilities that the first and second components meet specifications are 0.84 and 0.86. Assume that the components are independent. Determine the probability mass function of the number of components in the assembly that meet specifications. X = number of components that meet specifications. Round your answers to four decimal places (e.g. 98.7654).

Answer 1

P(X=0) = 0.0224, P(X=1) = 0.2552, P(X=2) = 0.7224

Explanation:

Let's define the following events

A: the first component meet specification

B: the second component meet specification

`A^(c)`: the first component does not meet specification

`B^(c)`: the second component does not meet specification

then

P(A) = 0.84, P(
`A^(c)`) = 0.16, P(B) = 0.86 and P(
`B^(c)`) = 0.14

we know that X = number of components that meet specifications, then, X can take the values 0, 1 or 2. Let's compute the probabilities P(X=0), P(X=1), P(X=2).

P(X=0) = P(
`A^(c)\cap B^(c)`) = P(
`A^(c)`)P(
`B^(c)`) (because components are independent)

= (0.16)(0.14)

= 0.0224

P(X=1) = P[
`(A\cap B^(c))\cup(A^(c)\cap B)`]

= P(
`(A\cap B^(c))`) + P(
`(A^(c)\cap B)`) (because
`(A\cap B^(c))` and
`(A^(c)\cap B)` are mutually exclusive)

= P(A)P(
`B^(c)`) + P(
`A^(c)`)P(B) (by independence)

= (0.84)(0.14) + (0.16)(0.86) = 0.2552

P(X=2) = P(A∩B)=P(A)P(B) (by independence)

= (0.84)(0.86)

= 0.7224

and the probability mass function of X is given by

P(X=0) = 0.0224, P(X=1) = 0.2552, P(X=2) = 0.7224