# A Happy Probability Question

Today in a literature class, we were given a 28 question, machine-graded, multiple choice, single correct answer test.
All questions had 4 possible choices.
We were instructed to answer only 25 out of the 28, as the teacher would only be grading out of 25, but wanted to give us a break.
Bull-hockey, said I. I’ll answer all 28, most likely get at least 3 wrong, and let the machine decide which three I skipped.
No no, said the prof. You must answer only 25. Otherwise you increase your chances of getting a good grade, and that wouldn’t be fair to the other students.
So, the queston: What is the difference in probabilities (of a perfect score, say) between these two systems? Also, in the average case, what is the difference in score? Is this a solvable problem?

The answer depends on a number of things. For example, it depends on how much you know about literature (i.e., how likely you are to answer a given question correctly). It also depends on how your quiz is graded if you do decide to answer all 28 questions.

Let’s assume for the moment that your odds of answering each of the 28 questions is equal (it’s an unrealistic assumption, but bear with me). Then if the grader just adds up the number of correct answers, and lowers the score to 25 if it was greater than 25, then yes you’d do better to answer all 28.

You can’t work out an exact numerical probability of a perfect score without more data or more assumptions, I’m afraid. In particular you need to know your odds of answering each particular question correctly (25%? 90%?).

If you choose the 25 questions to do “randomly” and
then select one of the 4 answers for each one “randomly”,
the probability of a perfect score is
(1/4)^25.
If you select one of the 4 answers for each of 28 questions
“randomly”, then the probability that your score
is greater than or equal to 25 is
(14926)(1/4)^25(3/4)^3 + (1427)(1/4)^26*(3/4)^2

• 28*(1/4)^27*(3/4) + (1/4)^28
Since (14926)*(3/4)^3>1, the second probability is
clearly larger.

Pretty tough to answer and not enough data.

If one assumes that there are at least 3 questions that you have no idea about with 3 choices each then chances are pretty good (80%) that you would get at least one extra mark just by guessing at the answers rather than not answering the toughest 3. Usually with multiple guess you can eliminate at least one wrong answer so that would increase the odds accordingly.

It depends very heavily on how the machine chooses the ones you skipped, as well as what assumptions we make about your probabilities of answering them correctly. If it will be kind enough to decide that you skipped 3 wrong answers, you are gaining an advantage over playing fair, given that you actually guessed on more than three questions, as suggested by your prof. If it will simply randomly pick 3 to throw out, it could be to your disadvantage, as it could throw out some of your known correct answers.
If we assume that you purely guess on every question, and it throws out three wrong:

you play fair - you answer 25, expect to get 6.25 correct.

you answer 28 - you expect to get 7 correct, it will throw out 3 wrong ones, still leaving you with 7 correct. Not fair.

Hmm…I should have known that would be important.
Using my previous test scores, I find I have a 76% chance of getting any single question correct.
Also, based on previous experience, the grader will count the test out of 25, no matter how many questions were answered.
From the random case calculations curiousgeorgeordeadcat gave, I can see I’m in some trouble
Now I’m just wondering how much…
Thanks,
plav

Oh, and if you aren’t guessing one of 4 answers
“completely randomly”, but it is reasonable to assume
that your probability of getting any given question
correct is the same probability p, then you can
replace 1/4 by p above and 3/4 by 1-p. Then it
isn’t so obvious the second probability is larger.
But it is, since for the second method,
P(>=25 correct)>P(first 25 correct, any performance
on last 3)=p^25.

I just recalled you asked for the average scores.
With probability p of getting each question correct,
assuming your performance on one question is independent
of your performance on another, and assuming you are
equally likely to pick any subset of 25 questions from
the 28 for method one, your average scores are:
For method 1 (answering only 25):
25p
For method 2 (answering all 28, getting the smaller