[Quantumcat]

Restricted choice can help you in multiple-choice exams.

For instance, say a question has options A, B, C which, using your knowledge of the subject being examined, you assign probabilities of being correct of A: 45%, B: 35% and C: 20%. So you of course choose answer A.

Then, later on, you read a question which makes it obvious that answer B (in the question under consideration) is incorrect. Should you switch your answer to C?

Most people would scoff and say of course not, we thought it was probably the wrongest answer (we assigned it a 20% chance of being right).

However, restricted choice says you should: C now has 55% chance of being right.

Why?

When you chose answer A, you only had a 45% chance of being right - that is, if you could somehow hedge your bets and choose B AND C [winning when either was right], you would have a 55% chance of being right. Knowing B is wrong doesn't change that, only now you have a legitimate way of being 55% right - switching your answer to C.

It gets even better if you chose your first answer at random - now you have a 66% chance of being right!

I just thought of this this afternoon. Hopefully this might come in handy for other students!

[Cascade]

I think that your new information just tells your that your initial subjective probabilities were incorrect.

You should reassess the probabilities based on the new information but not necessarily switch.

[masse24]

I would suggest studying hard so you don't need to guess...

[Antrax]

I think you mean this: https://en.wikipedia...ty_Hall_problem

[gnasher]

Quantumcat, on 2012-March-18, 21:47, said:

When you chose answer A, you only had a 45% chance of being right - that is, if you could somehow hedge your bets and choose B AND C [winning when either was right], you would have a 55% chance of being right. Knowing B is wrong doesn't change that, only now you have a legitimate way of being 55% right - switching your answer to C.

I'm going to pretend I hedged my bets by choosing "A AND B" (though I would call it "A OR B"), which would have given me an 80% chance of being right.

[mycroft]

The difference between this and the Monty Hall program is the key choice that Monty has to show you a losing option, and always knows which losing option is available. If you had picked B, he would have shown you the incorrect choice in A or C. It just turns out that you didn't pick B, and it happens to be wrong.

How much that effects the math, of course, is beyond my math (at least on a day I have to do a lot of other work).

[awm]

This is not true.

The essential ingredient in restricted choice is that the selection of the wrong answer you are warned about depends on your choice and is randomly selected.

Here, you were always going to be shown B as a wrong answer even if you selected B so it does not apply.

[jdeegan]

Not the Monte Hall problem. You are just reassigning probabilities based on a little more reflection. For 'Monte Hall' you need another independent actor, say ...... Monte Hall or the bridge opponent on your right (or left).

[Statto]

If set properly, the multiple choice answers should be completely randomized, so you can't draw any inferences.

[Quantumcat]

The answers need some thought behind them - each answer has to be plausible drawing on a particular misconception the student might have in the subject. They can't really be random. Although I'm not sure what you mean by random. Is it writing a bunch of words to do with the subject, and drawing two out and having them as the wrong answers?

But anyway the person who pointed out that the answer we find out to be wrong is random and not based on the knowledge of which one you picked has convinced me that I was talking rubbish!

[Statto]

I mean that you can't deduce whether A, B, C or D is more likely to be right based on previous questions.

[helene_t]

The Monty Hall problem would be like this:
- The student guesses (in this case A because it has 45% probability) and tells the teacher that A was his initial choice.
- The teacher must now, according to the conditions of the exam, tell the student that a particular one of the remaining two options is incorrect. The teacher will never tell the student whether his initial choice was correct.
- The student now can switch or not.

If this were the conditions of the exam then you are correct.

However, if the student just randomly learned about one alternative being incorrect, she should stick to her original choice, option A.

The difference is this:
- if she might learn about any alternative, including her original choice (the teacher might have told her that option A was incorrect), the MH problem (restricted choice) does not apply
- if she, according to the rule, can learn only about one of the two other options she did not originally choose (B or C), then restricted choice does apply.

[EricK]

You are prepared to keep A at 45% and (B or C) at 55%. But by symmetry, you could equally have decided to keep C at 20% and (A or B) at 80%. This would imply that not swapping is by far the better approach.

Since these two seemingly equivalent approaches lead to different outcomes they must both be wrong.