# Decision Tree Problem: Evaluate probabilities and determine in terms of C, all the optimal decisions.

I'm struggling with this decision tree question:

A part of an aircraft engine can be given a test before installation. The test has only a 75 % chance of revealing a defect if it is present, and the same chance of passing a sound (good) part. Whether or not the part has been tested it may undergo an expensive rework operation which is certain to produce a part free from defects. The cost of rework is £1000. If a defective part is installed in the engine the loss is £5000. Suppose 1 in 8 of parts are initially defective, and the cost of the test is £C. (a) Draw the decision tree, evaluate all probabilities and determine, in terms of C, all the optimal decisions. (b) How much should the manager be willing to pay for the test?

I know it's probably a simple question, and I understand the basics, but I'm having trouble processing the wording, and what I have so far doesn't feel like it's the right answer. I have attached what I've done so far(sorry it's messy, as it was rough work), if someone could give me some help as where I have gone wrong that would be great. Thank you.

• Welcome to MSE. My compliments on a good first question. You explained the problem well, and showed your efforts, which were substantial. Keep it up. Feb 18, 2020 at 19:56

Referring to the bottom diagram, the first two branches look right (it cost $$625$$ to install the part without testing, and $$1000$$ to test it,) but the third branch doesn't look correct to me.

If you test, there are four possibilities:

• Part is good, passes test (GP)
• Part is good, fails test (GF)
• Part is bad, passes test (BP)
• Part is bad, fails test (BF)

The probabilities of these scenarios are

• GP:$$\frac{21}{32}$$
• GF:$$\frac{7}{32}$$
• BP:$$\frac{1}{32}$$
• BF:$$\frac{3}{32}$$

So, if a part passes the test, the probability that it is good is $$\frac{21}{22}$$ and if it fails the test, the probability that it is bad is $$\frac{3}{10}$$.

In the first case, it's obviously better to install the part than to rework it. If it was better to install it when the probability of being good was $$\frac78$$, it must also be better when that probability has increased. When the part fails the test, analysis similar to what you did earlier shows that it's better to rework the part.

Can you continue with the problem now?

One thing I should mention is that I haven't considered the possibility of retesting a part that fails. I'm not sure if that is supposed to be an option. You might want to investigate this.

• I thought I understood what you wrote, but I'm trying to draw my decision tree and I can't seem to make it make sense, because of the way the question is asked I assume that I should come to the conclusion that the chance node(the circle) for the test should be a smaller number than the chance node for the install part. I don't know if I'm making any sense, but I've seen a similar question and the answer for the optimal decision should be say a+C(cost)>/<625, but at the moment my chance node for the test(a) always seems to be larger than 625, which doesn't make sense if I want to say: a+C<625 Feb 18, 2020 at 22:23
• I'm not sure if my next branch after 'test' should be the four possibilities you stated, or just two branches with 'passes test' and 'fails test'. I've tried both and they give me a number larger than 625, so if my assumption above was correct I'm doing something wrong. Feb 18, 2020 at 22:30
• I changed something and the answer makes more sense, this is my new branch for the 'test' imgur.com/yrgw0wG , hopefully it's correct I'm not too sure but makes more sense than my other attempts. Feb 18, 2020 at 23:00
• I would say the branches after "test" should be "pass" and "fail". When you have to make the decision on what to do next, all you know is whether the part passed or not. Feb 18, 2020 at 23:33