Question on Bayes' Theorem Problem

I have a question on a probability question that I have been staring at for the last hour. Here’s the question:

A merger arbitrage hedge fund manager is considering using a scoring model to test the likelihood that a merger deal will close within six months. Back‐testing the model produced the following results:

  • P(Closed on time) = 0.70
  • P(Passed test) = 0.76
  • P(Passed test | Closed on time) = 0.82

The manager uses the model to score a new deal. If the deal fails the scoring test, what is the updated probability that it will not close within six months?

Now, The answer is given as 0.475. and the solution is provided as:

P(no close | failed) = P(no close)(P(failed | noclose) / P(failed) = 0.30 (0.38 / .24) = 0.475

I understand where all of those numbers come from except the 0.38. I feel like this is something simple that I’m overcomplicating. I understand Bayes’ Theorem but will admit that probabilities are one of my weaker points on mocks.

I believe that it is a whole lot easier to see if you draw a binomial tree. The first branch is Closed on Time (up) vs. Didn’t Close on Time (down). The second branches are Passed Test (up) vs. Failed Test (down).

There are four ending nodes:

  1. A deal closed on time and passed the test
  2. A deal closed on time and failed the test
  3. A deal didn’t close on time and passed the test
  4. A deal didn’t close on time and failed the test

The respective probabilities are:

  1. 0.70 × 0.82 = 0.574
  2. 0.70 × 0.18 = 0.126
  3. 0.76 − 0.574 = 0.186 = 0.30 × P(pass | didn’t close on time)
  4. 0.30 − 0.186 = 0.114 = 0.30 × P(fail | didn’t close on time)

From this you can see that P(fail | didn’t close on time) = 0.114 / 0.30 = 0.38. However, with the tree, you don’t need this number.

The answer you want has failing the test given, so you’re at node 2 or node 4. You want the probability that you’re at node 4, which is:

0.114 / (0.126 + 0.114) = 0.475.

Much easier once visualized. I had started drawing a tree but got stuck on node 3 & 4. Thank you for the explanation!

My pleasure.