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ratio and number of samples

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I have a table full of heads

Each head took a test and the only thing I know about the test of each head are their results

The results show the total number of questions and the number of correct answers 

Each head has a different total number of questions

I have to select the most intelligent head

 

Q=total number of questions

A=correct answers

 

one head took only 1 question and got it correct, other head took 4 and 3 were correct, is second head more intelligent than first?

 

what can I do?

 

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Let's imagine each head has a true probability of getting a question right, but this probability is hidden to you. If you start with a prior that the probability is uniformly distributed between 0 and 1, after A correct answers and B incorrect answers the posterior distribution is a beta distribution with parameters alpha=A+1 and beta=B+1. Now we can easily compute the expected value of the hidden probability to be the mean of the posterior distribution, alpha/(alpha+beta) = (A+1)/(A+B+2).

So in your case, the first head gets a expected hidden probability of 2/3, while the other head's is 4/6, which is the same. However, looking at the mean only is a bit simplistic. The probability that the second head is more intelligent than the first head is actually 10/21 = .47619.

But of course, the answer would be different with a different prior distribution of hidden probabilities.

Is this homework?

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