老师能解释这题怎么算的嘛，图怎么画的？

问题如下：

Question

• Forty percent of the companies tested will go bankrupt within a year: P(non-survivor) = 0.40.

• Fifty-five percent of companies tested will pass: P(pass test) = 0.55.

• There is an eighty-five percent probability that a company will pass the test given that it survives a year: P(pass test | survivor) = 0.85.

Using the total probability rule, the probability that a company passed the test given that it goes bankrupt can be determined. The P(pass test | non-survivor) is closest to:

An analyst develops a set of criteria for evaluating distressed credits. Companies that do not receive a passing score are classed as likely to go bankrupt within the next year. The analyst concludes the following:

选项：

A.0.22.

B.0.35.

C.0.10.

解释：

Solution

C is correct. The total probability rule explains the unconditional probability of an event in terms of probabilities conditional on mutually exclusive and exhaustive scenarios, where: P(A) = P(A|S)P(S) + P(A|S^C)P(S^C).

Given that P(non-survivor) = 0.40, then P(survivor) = 1 - P(non-survivor) = 1 - 0.40 = 0.60. Accordingly,

P(pass test) = P(pass test|survivor)P(survivor) + P(pass test|non-survivor)P(non-survivor)

0.55 = 0.85(0.60) + P(pass test|non-survivor)(0.40)

Thus, P(pass test|non-survivor) = [0.55 -0.85(0.60)]/0.40 = 0.10.

A is incorrect because the total probability rule is a weighted average probability of all possible scenarios. This answer incorrectly applies the multiplication rule, which holds that the joint probability of two independent (not conditional) events equals the product of the two individual probabilities for non-survivors and passing the test: (0.40)(0.55) = 0.22, which does not account for all mutually exclusive and exhaustive scenarios as required by the total probability rule.

B is incorrect because 0.35 is the result if the probability of P(non-survivor) = 0.60. The question states that the probability of P(non-survivor) is 0.40, not 0.60. It is calculated as [0.55 -0.85(0.40)]/0.60 = 0.35.