Solve the problem. Express the answer as a percentage.
-$31 \%$ of the workers at Motor Works are female, while $37 \%$ of the workers at City Bank are female. If a worker is selected at random, what is the probability that the worker will be from Motor Works, given that the worker is female? Round your answer to the nearest tenth, if necessary.
A) $45.6 \%$
B) $54.4 \%$
C) $37 \%$
D) $31 \%$

Question

Quizplus · Accepted Answer

To find the probability that a randomly selected female worker is from Motor Works, use Bayes' theorem. Let $ P(M) $ be the probability of selecting a worker from Motor Works, and $ P(F|M) $ be the probability that a worker from Motor Works is female. Similarly, define $ P(C) $ and $ P(F|C) $ for City Bank. The probability is given by:  $$ P(M|F) = \frac{P(F|M) \cdot P(M)}{P(F|M) \cdot P(M) + P(F|C) \cdot P(C)} $$  Assuming equal number of workers in both companies, $ P(M) = P(C) = 0.5 $.  $$ P(M|F) = \frac{0.31 \cdot 0.5}{0.31 \cdot 0.5 + 0.37 \cdot 0.5} = \frac{0.155}{0.155 + 0.185} = \frac{0.155}{0.34} \approx 0.456 $$  Thus, the probability is approximately $ 45.6\% $.

Solve the Problem 31%31 \%31% Of the Workers at Motor Works Are Female, While

Solve the Problem $31 \%$ Of the Workers at Motor Works Are Female, While