Making a decision

Question

Like the title "Making a decision", I am standing for a problem. I've calculated two probabilities.

$$P(\text{Match}) = 0.24$$

$$P(¬\text{Match}) = 0.76$$

Now I've to make a decision.

Does anybody know how I can do this with mathematics? I've heard about utility functions or likelihood functions. But I don't know how to start.

Edit It's part of an artificial intelligence system. For example: I have a couple of sensors. They measure the enviroment about an object like a toy car with a trailer.

Match stands that the trailer is loaded.

¬Match stands that the trailer is not loaded.

The decision iam thinking of would tell me, 0.76 is enough to say that the trailer is loaded and the toy car can pass. Otherwise no pass. So I'm locking for an equation or a construct with a defined threshold to make this decision. Sorry if I can't describe it better, but english is not my native language

Greetings Jason

Answer 1

I think i have more problems with my "linguistic accuracy" :)

Thx Rick Decker for your answer. But I see that I must be a little closer. Another Example:

I am a customs officer patroling on a ship the sea border of England. I want to know if a ship is doing a bootleg or not. A couple of sensors giving me informations about incoming ships that want to land on the English Island. Lets call one ship "Bermuda".

The ship "Bermuda" has (through calculations) the probability to bootleg of 0.76 the complementary (not to bootleg) is 0.24

Now I want to make a decision. Can I trust the calculation with a probability of 0.76 (e.g. is the probability of 0.76 enough to trust my sensors so I stop them) or do I have to let them pass?

So I'm looking for a model or math construct that helps me with my decision. I was told to do this with utility function or likelihood. But I'm new to this field of mathmatics.

I hope this is a little bit more precisely.

Answer 2

For pure utility approach must look at the benefit of successful interception and/or the cost of failure. Put quite simply, if the cost if failure is $C$ then your expected cost is $C.P(fail)$. Here it becomes important as to the expected cargo of the bootlegger. If it's counterfeit Iron Maiden t-shirts, that would be considered quite a low cost if missed. If however it were guns, drugs and slaves, then the cost of failure might be considered quite high.

I think what you are also looking for is similar to a system I implemented some time ago whilst working in fraud detection. You have a resource issue. That is to say that your coastguard is a limited resource. If you set a threshold $T$ such that your coast investigates the boat if $P(bootleg) > T$ then if $T$ is too low, then the coastguard will be overloaded and may miss a boat. If $T$ is too high, then there is a chance that the coastguard will simply sit around doing nothing whilst bootleggers pass by. You must choose a threshold which maximises resource use and this can only be determined by the number of available coastguards, the number of boats and the amount of time it takes to investigate. It may be that you look at the problem from a "how many coastguards do we need" angle too as this could mean you could fix the threshold and staff appropriately.