Fuzzy probability/intuition problem

[jammieg]

A garaged door installer installs a door for Mrs. Banks one day and after completing the job she checks it out to find that it looks great except for a small dent on the metal panelling on the outside of the building, approximately 10 cm away from the garage door, she insists that the dent was not there this morning but is not certain. The door installer claim that he doesn't recall that happening and if it did he wasn't aware of it. She says she will ask her husband later if he made the dent or remebers it before the installer came.
Setting aside the motto the customer is always right here are some arbitrary facts:
The installer makes 1 mistake every 5 doors installed and of those 1 out of 10 he is unaware of the mistake.
Mrs. Banks is not sure of herself let's say she is 80% sure the installer did it and 20% it was caused by someone or something else.
Before Mr. Banks is consulted there is a 1 in 3 chance of he also being 80% sure the installer did it, but a 2 in 3 chance of being 80% sure he did not do it.
If the combined total sureness of the two are 80% or greater on some particular outcome they will believe the installer did it.

What are the overall odds that in the morning the installer will get a call from the Bankses saying they think the installer was responsible?

The answer is entirely up for debate this is an orginal problem.

 

[jammieg]

The solution I got was a 37% probability the Bankses will call.
Ok that one's way too easy, and I'm wrong anyway. What I meant was what is the probability the installer is guilty, oh well silly problem.

 

[tacman]

There is no clear answer to this problem as it is based on subjective and uncertain factors. However, based on the given information, we can make some assumptions and calculate a probability.

First, let's assume that the garage door installer did indeed make the dent on the metal paneling. In that case, the probability of Mrs. Banks being 80% sure that the installer did it is 80%. This is because she is already 80% sure that the installer did it and there is a 100% chance that she will believe it if her husband is also 80% sure.

Now, let's consider the possibility that the dent was caused by someone or something else. In this case, Mrs. Banks is only 20% sure that the installer did it. However, if her husband is also 80% sure, then the combined total sureness of the two is 84%. This is still above the threshold of 80%, so they will still believe the installer did it.

Based on these assumptions, we can say that the overall odds of the installer getting a call from the Bankses saying they think he was responsible is 80% (assuming that the dent was indeed caused by the installer).

However, it is important to note that this is just a hypothetical scenario and the actual odds may be different depending on the individual circumstances and factors involved. Additionally, the concept of "sureness" or "certainty" is subjective and can vary from person to person. Therefore, it is impossible to accurately determine the overall odds in this situation without more specific information.