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

Question

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.

Answer 1

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.