How does probability change as a group becomes less "average"?

Question

500 people are asked asked a question with a difficulty rating of 1 in 5. Meaning, the question is graded as being the type of question that, on average, only 1 out of 5 people asked are likely to get right.

Therefore, it can be assumed that around 100 people will answer the question correctly.

Those 100 people are then asked a new question with the same difficulty rating. (1 in 5)

How many of the 100 people (who answered the 1st question correctly) would be expected to answer the new 2nd question correctly? And how is this worked out?

I'm not sure whether or not this is a difficult question, it's certainly difficult for me as I am not very mathematic. It is something that has come up on a project I am working on, so if anyone can explain this to me in a simple way, I would be very grateful.

Thanks Tim

Answer 1

To conclude that 20 people in each of the original five groups will pick the correct answer out of five possibilities given, you need to assume that they are choosing their answers uniformly randomly. This is possibly not a realistic assumption depending on the nature of the question - even if it's not something for which most people know the correct answer, it might turn out that, e.g. one answer is obviously wrong. In that case, you might have $25\%$ of the group getting the answer right, because they're guessing randomly from the four remaining answers.

What you should expect of the 6th group hinges on whether you make this assumption. If you do, the answer is exactly the same as with the original groups - $20$ people will be expected to answer (at random) correctly, because their answer in the second question is completely independent from their first answer.

However, in practice and intuitively, people aren't expected to choose their answers uniformly randomly unless they don't have any idea, and people who do know the answer to one question are expected to have a better-than-random chance of knowing the answer to another. (i.e. the answers are not independent.) This means that we should expect more than $20$ people in the 6th group to answer correctly, but how much more depends on how likely a given person is to know the answer to the first question, and of the people who do, what proportion of them know the answer to the second. (Which is likely to be different than the proportion of all the people who know the answer to the second question, because "all people" includes the people who know neither answer)

Answer 2

If you assume that people can answer these questions, whatever they are, with a proportion $\frac15$, then after the first question $100$ remain and from these $20$ pass the second.

But if the second question is in the same area of competence as the first, odds are high than more than $20$ will succeed. (In the extreme, if the second question is the same as the first, the $100$ answerers will remain.)

This tells you the difference between independent and dependent drawings. Without more information, the question can't be answered.