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I want to find the probability that my student is a random guesser. On a 360-item multiple choice test with four choices for each question, he got 28.5% or 103 of the questions correctly.

Here is what I have so far. As everyone knows, the expected score is 25% or 90 items. Assuming that he is indeed a random guesser, I used the binomial distribution to get the variance np(1-p) = 360(.25)(.75) = 67.5; hence, a standard deviation of 8.22. Further assuming that random guessers are normally distributed, his z-score is (103-90)/8.22 = 1.58, making him an outlier. This places him in the top 6% of random guessers. This suggest that either (1) he is a very good guesser, (2) he is a very lucky guesser, or (3) he is NOT a random guesser at all.

Now I don't know what other concepts to use to find the probability that he is a random guesser. I don't even know if there is enough information; nor do I know whether all my computations and assumptions make any sense. I hope you can help. Cheers!

PS: I only had a 3-unit statistics course way back in college. "Dummifying" your explanations would surely be appreciated. Cheers! :-)


Edit: Thanks for all your help. So I guess it's really not that easy to get a good approximation on the said probability.

Having said that, is there a relatively simple way to get even a very crude approximation of the answer? For instance, even before posting the question here, I actually considered the Bayesian probability mentioned above. To make things simple, I assumed that P(getting 103|guesser) is simply ${{360}\choose{103}}*.25^{103}*.75^{360-103}.$ And just to have a starting point, let's just say that 1 out of 5 students are random guessers, so P(guesser) is 0.2. What would be a reasonable initial estimate, albeit inaccurate, for P(getting 103)?

Then maybe we can play around with the assumed values later to get a spectrum of possibilities.

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    You should run your students through the same test again and again. Questions they know will receive constant answers, questions they guess won't :)2012-09-09
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    Which _alternative_ to being a "random guesser" are you considering? We can conclude definitely that the student is not someone who knows the correct answer to all questions. What, then, would you have him do with the rest of the questions _other_ than guess randomly? It does not make sense to assign a probability to "being a random guesser" unless you can suggest something _different_ that would also have positive probability.2012-09-09
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    There might be a 4th possibility: out of 360 questions, he correctly solved around 10 to 15 problems, and guessed the rest. It can be calculated approximately how many did he solve to achieve an expected score of 25% for the guesses.2012-09-09
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    It's not that it's hard to get a good approximation for the desired probability; the problem is that you haven't actually defined any probability. As Henning said, it all depends on the alternatives, and the difference they make isn't just an approximation error, it's the entire answer. If you assume that the proportion of answers the student knows is uniformly distributed between $0$ and $1$, then the probability that it's $0$ is $0$, whereas if you assume that it's either $0$ or $1$, then the probability that it's $0$ is $1$.2012-09-12
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    @Mark: I've merged your two accounts. Please consider [registering your account](http://math.stackexchange.com/users/login). This helps the software keeping track of you so that you retain the ability of commenting on this thread, editing your question, etc.2012-09-12
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    A true random can be on outside the probability range, because we often expect a random sequence to be uniformly distributed, but in real it can be nothing like you would expect. This is a very informal video that explains what I'm saying http://youtu.be/Lf4ZmWc_jmA?t=8s2013-03-22

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