Formula for Determining Expected Matches - Given #subjects, #options, P(match)

[JZMathTime]

Imagine a situation where a number of subjects (X) are going to be offered a number of options (Y), and there's a constant probability (P) that any given subject will accept any given offer.

An option that is accepted will be "filled" by exactly one of the subjects that accepted it. And while a subject may have been willing to accept multiple options, a subject can only fill exactly one option.

I'm trying to derive a formula for the expected result of this scenario, but getting stuck. This would be extremely easy to calculate if we were just looking for the expected number of matches overall, but the fact that we're actually "filling" the options as we go is throwing me for a loop.



In other words --
Given:
Subjects are presented with Options, which they may accept or not.
An option that is accepted by any number of subjects greater than 0, will be filled by exactly one of those subjects.
Each subject can fill at most one option, but may accept more than one.
Let X = Number of Subjects
Let Y = Number of Options
Let P = Probability of a Subject accepting an Option
Define F(X,Y,P) such that it gives the expected number of options that will be filled by the subjects.


This seems like such a basic question that there must already be a formula someone derived that describes these situations, but I haven't been able to search it up online. Can anyone offer any help here?

 

[Dr.Peterson]

Is a person selected for each option randomly from among those who accept it, or is there an attempt to maximize the number filled, by selecting people strategically? For example, if only one person accepts option A, but he would accept every option, does he have a 1/Y probability of being given option A, or probability 1? Or does each person not have a list of what he will accept, but instead randomly says yes or no as he is given an offer, with probability P each time? (That doesn't sound like "willingness" to me!)

I think you have to clarify the scenario. I can't tell what the sequence of events is, specifically whether a person gets a list of offers at once, or one offer at a time, and whether each option is offered to a list of people at once, or to one at a time.

 

[JZMathTime]

Is a person selected for each option randomly from among those who accept it, or is there an attempt to maximize the number filled, by selecting people strategically? For example, if only one person accepts option A, but he would accept every option, does he have a 1/Y probability of being given option A, or probability 1? Or does each person not have a list of what he will accept, but instead randomly says yes or no as he is given an offer, with probability P each time? (That doesn't sound like "willingness" to me!)

I think you have to clarify the scenario. I can't tell what the sequence of events is, specifically whether a person gets a list of offers at once, or one offer at a time, and whether each option is offered to a list of people at once, or to one at a time.

To clarify, offers are given to all subjects simultaneously, but only for one option at a time. If more than one subject accepts the offer, the assignment is determined randomly before moving on to the next option being offered to all un-assigned subjects.

Whether a subject will accept an offer has to do with whether or not they are a "match" for the option, the matching factor is assigned randomly to each subject and option at the start of the experiment, in such a way that there is probability P of alignment between a randomly chosen subject/option pair.

 

[Dr.Peterson]

So P is not a constant, but varies from subject to subject? That's a whole different kind of problem.

 

[JZMathTime]

So P is not a constant, but varies from subject to subject? That's a whole different kind of problem.

Sorry, P should definitely be the same for everyone, I must be explaining something wrong. Maybe I should have labeled it as "Let P = Expected Rate of Subject/Option matches" or something like that.

The idea is that if (for example) P were 10%, then we'd start the experiment by assigning each subject and each option a random number from 1 to 10. Subjects/Options that are assigned the same number are a "match"

That's all that's meant to be captured by that probability, the likelihood of a randomly chosen subject/option pair to be a match.

If the problem is solveable with setting that to a constant 10%, then that would be more than satisfactory, I just figured it should be derivable with the probability as a variable as well.