A good friend of mine, David Jenkins, once called me asking me a question I don't think I ever fully solved. He wanted to know the applicability of multiple choice questions to the Monty Hall problem. Update 12/21/21: looks like it's not applicable.

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Modified multiple choice test scenario:

Say you chose the correct one (1/4 chance), then eliminate one of the incorrect ones, then the switching method gives you 0 percent chance of being correct. 

Say you choose an incorrect one (3/4 chance), then eliminate one of the other incorrect ones (but not the one you chose), then switching gives you 1/2 chance of being correct. 

Overall chance of being correct is is (1/4 * 0) + (3/4 * 1/2) = 3/8 = 37.5%. This is higher than 1/3. In this scenario, you should always switch.

Actual multiple choice test scenario:

Say you choose the correct one (1/4 chance), then eliminate one of the incorrect ones, then the switching method gives you 0 percent chance of being correct. 

Say you choose an incorrect one (3/4 chance), then there are really two scenarios:

(a) You eliminate the one that you had chosen (1/3 chance of this happening), then you would definitely want to switch. Chance of being correct when you switch is 1/3.

(b) You eliminate one of the other incorrect ones (2/3 chance of this happening), then the switching method gives you a 1/2 chance of being correct.

Overall probability is: (1/4 * 0) + (3/4 * (1/3*1/3 + 2/3*1/2)) = (3/4 * 4/9) = 3/9 = 1/3