# How to calculate the best way to fill out this test?

1. Feb 4, 2014

25 questions, multiple choice, 5 choices per question.
Correct = 6 points
blank = 1.5 points
wrong = 0 points.

I should know how to do this, but have a cold and simply can't wrap my head around it…

Danke Schoen!

2. Feb 5, 2014

### Staff: Mentor

If you want to maximize the expected points, answer as soon as you can exclude 1 answer (then you get on average (0+0+0+6)/4=1.5 points, the same as leaving it blank).
If you want to maximize the probability of getting 100%, fill in everything (just as an example that there can be different strategies for different targets).

3. Feb 5, 2014

### bahamagreen

I think this is what you are looking for.
Working out the probable expectation points for the rest of the possible situations...

Looks like if you can identify and eliminate 1 or more answers as incorrect,
guessing among those remaining is as good or better than leaving it blank.

6 points for correct
same as guessing among 1 when all 4 wrong answers are eliminated

3 points for guessing among 2
1/2 x 6 = 3
this is guessing after eliminating 3 of the answers

2 points for guessing among 3
1/3 x 6 = 2
this is guessing after eliminating 2 of the answers

1.5 points for guessing among 4
1/4 x 6 = 1.5
this is guessing after eliminating 1 of the answers

1.5 points for leaving blank

1.2 points for guessing among 5
1/5 x 6 = 1.2
this is guessing without eliminating any answers

0 points for wrong

Get well soon...

edit... mfb, I'm unclear on the "...maximize the probability of getting 100%, fill in everything..." strategy. Leaving it blank is 1.5 points, but guessing without elimination is 1.2 points.

4. Feb 5, 2014

### Staff: Mentor

If you leave one blank, you cannot reach 100% any more.

In the same way, to optimize the probability to get at least 25%, don't fill in anything.

5. Feb 5, 2014

### bahamagreen

I'm was feeling a conflict between the strategy for making the highest possible score and the one you suggest for the best chance to make 100%... I think you are right, but it didn't feel right. :)

If you know the answer to 24 of 25, then you might justify "going for it" and guessing that one rather than leaving it blank. But the fewer you know you have right, the less likely this would work. The extreme case of going in clueless and not knowing any of the answers would require one to guess correctly all 25 times.

I think that would be p=(1/5)^25 or 1 out of almost 300 quadrillion.

I'm thinking that if you graphed both strategies (probability of scoring 100% over number of answers not known) that the max score strategy will start at 0% and remain flat because of choosing a blank over a total guess. The no-blanks strategy starts at .2 with one unknown answer, and approaches (1/5)^25 with subsequent unknown answers. So you are correct.

But, if you plot probable scores rather than probability of 100%, the max score strategy starts off at .25 and remains flat at .25, but the no-blanks method starts at .2 and fall fast.

If I did this is right (?), I'm surprised you are correct. I would not have seen it that way, Thanks.

6. Feb 5, 2014

### Staff: Mentor

If you just have "get 100% or fail", guessing all is still the best option, because leaving a question blank gives you a probability of zero.

To make a more realistic example: If the test is "get at least 40% or fail" and you have no idea about the answers, guessing all should be the best strategy (and it is clear that guessing some answers is necessary). You need 10 correct answers, with expected 5 - this gives a ~2% chance to pass.