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August 13

What is this relationship called?

Imagine some educational assessment with 5 items each with dichotomous outcomes. There is/are:

1 way to get 100%,
5 ways to get an 80%,
10 ways to get a 60%,
10 ways to get a 40%,
5 ways to get a 20%, and
1 way to get a 0%.

Combining these probabilities with the outcome utility in an expected utility, the expected utility:

for 5 correct answers is .03125,
for 4 correct answers is .125,
for 3 correct answers is .1875,
for 2 correct answers is .125,
for 1 correct answer is .125, and
for 0 correct answers is 0.

Moreover, .03125=.5^5, .125=(.5^4)*2, and .1875=(.5^3)+(.5^4).

Now, imagine a similar assessment with 3 items with polytomous outcomes, where there is a 25% probability of getting the answer correct (4 multiple choice answers, 1 is correct). There is/are:

1 way to get a 100%,
9 ways to get a 66%,
27 ways to get a 33%, and
27 ways to get a 0%.

Again, using the expected utility function, the expected utility:

for 3 correct answers is .015625,
for 2 correct answers is .09375,
for 1 correct answer is .140625, and
for 0 correct answers is 0.

Moreover, .01625=.25^3, .09375=(.25^2)+(.25^3)*2, and .140625=(.25^2)*2+.25^3.

Is this some fascinating relationship, or am I just playing with numbers? Schyler (exquirere bonum ipsum) 16:03, 13 August 2018 (UTC)[reply]

The number of correct answers is an example of a binomial random variable. In the first case, n = 5, and p = 1/2, and in the second, n = 3, and p = 1/4. –Deacon Vorbis (carbon • videos) 16:09, 13 August 2018 (UTC)[reply]

Great! What about the relationship between the expected utilities and the other parameters? Schyler (exquirere bonum ipsum) 16:12, 13 August 2018 (UTC)[reply]

I'm not sure what you mean by "utility" exactly, but from the article there, the expected number of correct answers is just np. –Deacon Vorbis (carbon • videos) 16:13, 13 August 2018 (UTC)[reply]

The percent correct is the utility, so the expected utility is the probability of a correct answer times the percent correct. For example, the exepcted utility of getting four correct answers on assessment a is (5/32)*(0.8)=.125. Schyler (exquirere bonum ipsum) 16:21, 13 August 2018 (UTC)[reply]

Expected has a technical meaning (see Expected value), which I mention because it doesn't make sense to talk about the "expected ___" of getting 4 answers right out of 5. There's also a meaning of utility in economics, which may be what you're thinking of here, but I'm not sure. You seem to be wanting to take the expected value of a function of a random variable, where the function is just dividing by the number of questions, in order to convert from a raw score to a fraction. By the linearity of expectation, it doesn't matter what order you do that in. So you can just find the expected number of correct answers (which was np as noted above), and then divide by the total number of questions (n), to get the expected score (p). This just verifies the intuitive result that you if you have a probability p of getting each question correct, then you should expect to get a total score of p on the test as well. –Deacon Vorbis (carbon • videos) 16:31, 13 August 2018 (UTC)[reply]

I am wondering about the distribution of expected utilities in the econometric sense. Sorry if that was a problem. The expected utility is n_correct*p(n_correct), not np. I am looking for the probability density function of the expected utility, where the expected utility is a function of n_correct and p(n_correct). Then, the 'expected value' is of less interest than the 'maximum,' although I think they may be the same thing in the end. I want the maximum because I want to maximize my expected utility when taking an assessment, for example. Schyler (exquirere bonum ipsum) 16:45, 13 August 2018 (UTC)[reply]

You don't really have an expected value of a single outcome. Expected value is an average (technically, an integral with respect to a probability measure) over all possible outcomes. The only difference you seem to be talking about is fractional score versus number of questions correct, but that's a fairly trivial difference. Moreover, there doesn't seem to be anything to maximize; you just answer what you think has the highest chance of being correct (if you're just guessing, then it doesn't matter), and the expected score is as I noted above. –Deacon Vorbis (carbon • videos) 16:54, 13 August 2018 (UTC)[reply]

It is not clear to me exactly what the "fascinating relationship" is supposed to be, but to a first approximation it looks like Pascal's identity (which is related to the fact that you're talking about a binomial distribution). Not sure why the conversation is getting hung up on the uninteresting terminological question, but the way that I would phrase it is "the contribution to the expectation coming from k right answers is ..." or something like that. (If you want to maximize your outcome, you should try to get the largest number of questions right; this has nothing to do with the distribution you're looking at.) --JBL (talk) 23:27, 14 August 2018 (UTC)[reply]