# The odds of randomly passing Multiple Choices questions?

I hope this is the right place to ask this question as it is related to probability and stats.

Basically, we were supposed to have final exams in the form of a written one, yet due to the pandemic, it was changed to a remote multiple-choice test. The test had 30 questions, each question got 4 choices (only 1 correct answer). Choosing the wrong answer will cause a student 1/3 mark deduction. The thing is, once a student tick to any answer, they aren't allowed to untick to make that question become blank to avoid a possible 1/3 mark losing.

To me, I think it was an issue. But when I complained about this issue with the guy who's been in charge, he responded to me that the setup of not letting student uncheck their answers, on average, will not disadvantage students. He also said that statistically, choosing not to answer or choosing to give random answers would make no difference as the average total score would still be zero.

I found it not correct. I think the situation he explained to me only correct when the number of questions students randomly choose is a big number. In that case, choosing randomly answers or leaving the answers blank will be as he claimed. But in the real exam, students often had a small number of questions they were not sure about, like 5 or 6 questions, and randomly choosing with 1/3 mark penalty for the wrong answer would put students in the disadvantage than otherwise.

• If you want to 'untick' only when--presumably after some informed reflection--you fear you marked the wrong answer, then it clearly makes a difference. (But then could you change to a 'better' answer?) The score averages 0 when a robot picks an answer or no answer at random – BruceET Jun 4 '20 at 21:57

To think about it another way - consider a game where we roll a 4-sided die, and I pay you \$10 on a 4, but you pay me \$3.33 on a 1-3. The scoring system on the test assumes that you should be indifferent to whether you play this game or not, but depending on your risk-taking aversion, you may just prefer to not play and just stick with your \$0. The variance of the outcome is higher if you choose to play one round (you'll either wind up with \$10 or -\$3.33), but the expected value of the outcome is identical (\$0).