Question

ii) Suppose 70% of all companies are classified as small companies and the rest as large companies. Suppose further, 82% of large companies provide training to employees, but only 18% of small companies provide training. A company is randomly selected without knowing if it is a large or small company; however, it is determined that the company provides training to employees. What are the prior probabilities that the company is a large company or a small company? What are the revised probabilities that the company is large or small? Based on your analysis, what is the overall percentage of companies that offer training?

Answer 1

The prior probabilities that the company is a large company or a small company are 0.3 and 0.7.

The revised probabilities that the company is large or small are:

$P(L|T) = 0.3×0.82/(0.3×0.82 + 0.7×0.18) = 0.6613.$

$P(S|T) = 0.7×0.18/(0.3×0.82 + 0.7×0.18) = 0.3387.$

The overall percentage of companies that offer training is:

$P(T) = 0.3×0.82 + 0.7×0.18 = 0.372.$