bay‘s formula

问题如下：

An auditor has developed a test to identify financial fraud in companies. Research shows that 90% of companies involved in financial fraud fail the test, while 95% of companies not involved in financial fraud pass the test. It is estimated that 8% of companies are involved in financial fraud. If a company fails the test, what is the posterior probability that it is involved in financial fraud?

选项：

A.

61.02%

B.

90%

C.

92%

解释：

We are given: P(A) = 0.08 (the probability of a randomly selected company being involved in financial fraud)

P(B|A) = 0.90 (the probability of a company failing the test given its involvement in financial fraud)

P(not A) = 1 - P(A) = 1 - 0.08 = 0.92 (the probability of a randomly selected company not being involved in financial fraud)

P(B|not A) = 0.05 (the probability of a company failing the test given its not being involved in financial fraud)

We need to find P(A|B), the posterior probability that a company is involved in financial fraud given that it fails the test.

By applying Bayes’ theorem:

P(A|B) = (P(B|A) * P(A)) / P(B)

P(B) can be calculated using the law of total probability:

P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)

Substituting the given values: P(B) = (0.90 * 0.08) + (0.05 * 0.92) = 0.072 + 0.046 = 0.118

Now, calculating P(A|B) using Bayes’ theorem:

P(A|B) = (P(B|A) * P(A)) / P(B) = (0.90 * 0.08) / 0.118 = 0.072 / 0.118 ≈ 0.6102

The posterior probability that a company is involved in financial fraud given that it fails the test is approximately 61.02%