MULTIPLE Choice (part 1)
Usually multiple choice problems are actually "singular", since you are only choosing one out of 4 or 5 possible choices.
What if you can choose more than one choices?
A problem has 4 possible choices (A, B, C and D). The correct answer is a nonempty subset of the set of all choices (like A, A&D, or even A&B&C&D). 2 points will be given if your subset is same as the correct answer, and 1 point will be given if your subset is also a subset of the correct answer. No points will be given otherwise, but there's no penalty as well (yay for guessing).
Now you actually confronts such a problem, and unfortunately you are not brilliant enough to figure out the answer (actually you have no idea). You know that every nonempty subset will have equal chance for being the correct answer, and you don't want to submit nothing (an empty set) since no points will be awarded for that. However, you are brilliant enough to figure out how to maximize your expected number of points on this problem. What is this expected number?
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Since we have no idea what the answer is, and all answers are equally likely, WLOG we can either select 1, 2, 3, or 4 answers as our answer. In each case, we have 1 5 1 chance of getting 2 points, plus some additional chance that our solution is the subset of the correct answer, giving us 1 point.
This probability is clearly maximized when we only choose 1 letter. So our expected score is:
2 ⋅ 1 5 1 + 1 5 2 3 − 1 = 1 5 9 = 5 3