Prob tree for this problem

"15. You have developed a set of criteria for evaluating distressed credits. Companies that do not receive a passing score are classed as likely to go bankrupt within 12 months. You gathered the following information when validating the criteria: ● Forty percent of the companies to which the test is administered will go bankrupt within 12 months: P(nonsurvivor) = 0.40. ● Fifty-five percent of the companies to which the test is administered pass it: P(pass test) = 0.55. ● The probability that a company will pass the test given that it will subsequently survive 12 months, is 0.85: P(pass test | survivor) = 0.85. A. What is P(pass test | nonsurvivor)? B. Using Bayes’ formula, calculate the probability that a company is a survivor, given that it passes the test; that is, calculate P(survivor | pass test). C. What is the probability that a company is a nonsurvivor, given that it fails the test? D. Is the test effective? I cannot understand how to make a tree for this problem. The question is not clear. I am unable to figure out what comes first- survivor, non survivor or pass/fail. Is the question construction proper? Can we expect such a question on the exam?

P(A) = P(A&B) + P(A&~B) = P(A|B)P(B) + P(A|~B)P(~B)

P(pass test) = P(pass test|survivor)P(survivor) + P(pass test|nonsurvivor)P(nonsurvivor)

0.55 = 0.85 × 0.60 + P(pass test|nonsurvivor) × 0.40

P(pass test|nonsurvivor) = (0.55 – 0.85 × 0.60) / 0.40 = 0.10

P(survivor|pass test) = P(pass test|survivor)P(survivor)/P(pass test)

= 0.85 × 0.60 / 0.55 = 0.9273

P(nonsurvivor|fail test) = P(fail test|nonsurvivor)P(nonsurvivor)/P(fail test)

[1 – P(pass test|nonsurvivor)] × 0.40 / 0.45

= 0.90 × 0.40 / 0.45

=0.8000

Seems pretty good.

Given the data, it will easier to construct the tree with survivor/nonsurvivor first, then pass/fail second.

Yup.

Perhaps one of these questions.

No. This was not what I was asking for. Sorry, I couldn’t put myself across. All I wanted to know is how will aI create the probability tree in this case ( the two case tree which has been covered at Level 1). I was of the opinion that we could solve all such problems by constructing a tree, just that I couldn’t build one for this.

Well, then . . . start with:

Survive/Don’t Survive

  • Up, Survive, P = 1 – 0.40 = 0.60

    • Up, Pass, P = 0.85, Total P = 0.60 × 0.85 = 0.51
    • Down, Fail, P = 0.15, Total P = 0.60 × 0.15 = 0.09
  • Down, Don’t Survive, P = 0.40

    • Up, Pass, P = _ A _, Total P = 0.55 – 0.51 = 0.04
    • Down, Fail, P = _ B _, Total P = 0.40 – 0.04 = 0.36

You have to solve for A and B :

0.40 × A = 0.04

A = 0.04 ÷ 0.40 = 0.10

0.40 × B = 0.36

B = 0.36 ÷ 0.40 = 0.90

x = (survivor | pass test)= 51

| pass = 55

| (55-x) = (non-survivor | pass test) = 4

100 tests | ----------------------------------------------------------

y = (survivor | fail test) = 9

| fail = 45

(45-x) = (non-survivor | fail test) = 36

Given p(survivor) = 1-0.4=0.6 therefore total 60 survivors from 100 test takers

P(pass test | survivor) = 0.85 implies (x/60)=0.85 therefore x=51; y=(60-51)=9

A. What is P(pass test | nonsurvivor)? = 4/(4+36) = 0.10 B. Using Bayes’ formula, calculate the probability that a company is a survivor, given that it passes the test; that is, calculate P(survivor | pass test).=51/55= 0.93 C. What is the probability that a company is a nonsurvivor, given that it fails the test? = 36/45 = 0.80

I got confused because I was wondering as to how can something not survive and still pass? Weird. Anyway, thanks to both of you.