Question

Evaluating the performance of quality inspectors. The performance of quality inspectors affects both the quality of outgoing products and the cost of the products. A product that passes inspection is assumed to meet quality standards; a product that fails inspection may be reworked, scrapped, or reinspected. Quality engineers at an electric company evaluated performances of inspectors in judging the quality of solder joints by comparing each inspector’s classifications of a set of 153 joints with the consensus evaluation of a panel of experts. The results for a particular inspector are shown in the table. One of the 153 solder joints was selected at random.

Committee’s judgment joint	Joint Acceptable	joint Rejectable
Joint Acceptable	101	10
joint Rejectable	23	19

a. What is the probability that the inspector judged the joint to be acceptable? That the committee judged the joint to be acceptable?

b. What is the probability that both the inspector and the committee judged the joint to be acceptable? That neither judged the joint to be acceptable?

c. What is the probability that the inspector and the committee disagreed? Agreed?

Short answer

1. The probability is 0.81 and 0.73.
2. The probability is 0.12 and 0.66.
3. The probability that the inspector and the committee disagreed is 0.78.

Step-by-step solution

Important formula

The formula for probability are$\mathbf{P}\mathbf{=}\frac{\mathrm{favourable}\mathrm{outcomes}}{\mathrm{total}\mathrm{outcomes}}$

Find the probability that the inspector judged the joint to be acceptable.

Here, IA=the inspector judged the joint to be acceptable.

IR= the inspector judged the joint to be rejectable.

CA=the committee judged the joint to be acceptable.

CR= the committee judged the joint to be rejectable.

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\mathbf{IA}\mathbf{\right)}\mathbf{=}\frac{101+23}{153}\\ \mathbf{=}\frac{124}{153}\\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{81}\end{array}$

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\mathbf{CA}\mathbf{\right)}\mathbf{=}\frac{101+10}{153}\\ \mathbf{=}\frac{111}{153}\\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{73}\end{array}$

Thus, the probability is 0.81 and 0.73.

Determine the probability that both the inspector and the committee judged the joint to be acceptable. 

$\begin{array}{c}\mathbf{P}\mathbf{(}\mathbf{IR}\cap \mathbf{CR}\mathbf{)}\mathbf{=}\frac{19}{153}\\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{12}\end{array}$

$\begin{array}{c}\mathbf{P}\mathbf{(}\mathbf{IA}\cap \mathbf{CA}\mathbf{)}\mathbf{=}\frac{101}{153}\\ \mathbf{=}0.66\end{array}$

Hence, the probability is 0.12 and 0.66.

Find the probability that the inspector and the committee disagreed.

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\mathbf{AGREE}\mathbf{\right)}\mathbf{=}\mathbf{P}\mathbf{(}\mathbf{IR}\cap \mathbf{CR}\mathbf{)}+\mathbf{P}\mathbf{(}\mathbf{IA}\cap \mathbf{CA}\mathbf{)}\\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{66}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{12}\\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{78}\end{array}$

Therefore, the probability that the inspector and the committee disagreed is 0.78.