# A Strangely Familiar Probability Problem

You're taking the final exam for your Introduction to Probability and Statistics course, and once again you've failed to study (you were up all night solving online brainteasers).

The exam is multiple choice. It asks 120 questions, each of which has possible answers A, B, and C. You haven't the faintest idea how to answer any of the questions and so you turn to your old standby strategy: always answer B.

As quickly as you can, you fill out all the B's on your exam paper.

Shortly after you've finished, your professor gets up at the front of the class and announces that an error was made while printing the exam: answer C should not appear on the answer sheet because C is not the correct answer to any question.

You now have several options:

3. switch some, but not all, of your answers from B to A

Given your goal is to get as many answers correct as possible, what strategy do you use, and how many correct answers do you expect to get (on average) when using this strategy?

Puzzlers are politely encouraged to place answers in spoiler blocks to avoid inadvertently spoiling the fun for other readers. :)

• Sorry if this seems nit-picky, but the first sentence says I'm writing the final exam (so I'm the professor). But a moment later, I'm the student. Should the question start with "You're taking the final exam..."?
– HTG
Commented Nov 6, 2014 at 13:47
• @HTG You must be an educator. ;) It wouldn't have occurred to me that "writing an exam" would refer to the producing rather than the taking of the exam. I'll amend the language on your recommendation.
– COTO
Commented Nov 6, 2014 at 14:37
• Guilty as charged. I teach math at RIT (though not prob and stats).
– HTG
Commented Nov 6, 2014 at 16:09
• Of course, if the professor had announced that B is not the correct answer to any question, then your best strategy is to switch..but that's not Monty Hall either :)
– user4603
Commented Nov 6, 2014 at 20:59
• @HTG, I think students "write an exam" in Canada, "take an exam" in the US, and "sit an exam" in the UK, thus the confusion. In the US, the professor "writes the exam", which I understand is called "setting the exam" in other places. Commented Nov 6, 2014 at 23:30

Familiar indeed.

From the name and setup, it sounds like this was meant to invoke thoughts of Monty Hall type puzzles. However here what the professor revealed didn't depend on your choice, so is not actually like the Monty Hall puzzle. With absolutely no information about the distribution of answers besides that the probability of C is zero, guessing A is as good as guessing B. There is no reason to change your answer.

• It's important to note the difference from the classic problem - good answer
– Joe
Commented Nov 6, 2014 at 8:20
• Came to the same conclusion straight away, but was so hoping to be wrong and that the answer would be some implementation of option 3 :P Commented Nov 7, 2014 at 1:15

The original probability of the "Always answer B" strategy would have given you

1/3rd probability of getting everything correct and 40 questions right on average.

However after the cancellation of C as an answer

for each question, now you have 50% chance of getting the answer right with your original strategy. Since neither you do not have any clue regarding the correct answers, switching would give you no obvious advantage, leaving the choice of A for all questions with equal probability. Any attempt to increase the probability by choosing A or B would also leave a risk of decreasing the correct answers. This is different from Monty Hall problem where the elimination took into account your original choice.

So going by the above logic

In the interest of time savings, you dont switch

• "Different from the [original] problem where the elimination did not take into account..." - other way round, the original did take your choice into account
– Joe
Commented Nov 6, 2014 at 8:47

The correct answer: If you can figure out how to switch some of the B's to A's such that your grade will improve, well, like I said, your grade will improve.

• This may be the truest statement I've read on the internet all day. Commented Nov 6, 2014 at 22:30
• You'd be surprised how many people mistake this problem for "that other one", especially if you add the clause "The professor observes your answer sheet as he walks to the front of the class. Once at the front, he announces..."
– COTO
Commented Nov 6, 2014 at 23:32