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Say a person is taking a quiz which is composed of 10 true-false questions. Which of these techniques has the highest likelihood to maximize your score:
1) Randomly answering true-false for every question
2) Simply circling all answers as True or all answers as false.
Is there a more interesting strategy that I haven't answered?

How does the probability change when the answers are evenly distributed vs. when they aren't. I feel that choice 2 has the highest probability of success, but I cannot mathematically express this, while others seem to think it doesn't matter, but they cannot mathematical

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If one knows a little, but not very much, one can take advantage of the nasty habit of test-makers to try to fool people. Look at the more plausible-sounding of the two answers, and choose the opposite. – André Nicolas Oct 24 '12 at 15:47
The trouble with @André's suggestion is that if you chance to actually learn some of the material being tested, the right answers will start sounding more plausible to you than the decoys. So you need not only to have a little knowledge, but also a solid understanding of where the limits of your knowledge are -- and by the time you have that you'll probably be able to pass the quiz honestly after all. – Henning Makholm Oct 24 '12 at 17:55
@HenningMakholm: The comment above was only a little tongue in cheek. Particularly in high school tests, with too many too easy questions multiple choice in too little time, a good test-taker will generally beat someone who thinks. – André Nicolas Oct 24 '12 at 18:00

The question as posed isn't mathematical but psychological – it's about the psychology of the people who design the test. If they tend to ask questions to which the answer is true, always circling true is the best strategy. The question becomes mathematical if you assume that they randomize the questions such that each question independently has a probability of $1/2$ of correctly being answered by "true". In this case, any strategy not based on knowledge of the correct answers is as good as any other and results in an average hit rate of $1/2$.

And yes, there is a more interesting strategy: Learning the correct answers.

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Interesting, I thought there would be more to it. It seems somewhat strange that it is so simple. Maybe that was just my hope of gaining some interesting insight into probability. – Michal Frystacky Oct 24 '12 at 15:44
@Michal: It's that simple because the randomization of the questions completely wipes out any effect that your strategy might have. Perhaps it's easier to think of it with the events reversed in time: First you guess according to some strategy; then the test designers randomly choose which questions to formulate which way around. Then it's obvious that they'll make half your answers come out right. There's no reason why the time reversal should change the result, since you don't know the results of their randomization in either case. – joriki Oct 24 '12 at 15:47
Thanks, that makes the understanding of this problem much more intuitive. I believe this cleared some confusion I carried when considering the problem. – Michal Frystacky Oct 24 '12 at 17:37

If the answers are evenly distributed (5 true and 5 false) then option 1 has about a 0.1% chance of maximising your score at 10 out of 10 while option 2 has no chance of this. Circling 5 as true at random and 5 as false would have about a 0.4% chance of getting 10 out of 10 correct. These calculations are $2^{-10}$, $0$, and $1/{10\choose 5}$.

But all three strategies have an expected score of 5 out of 10, just with different dispersions. As joriki says, learning the subject of the quiz may help bias the expected score upwards.

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Thank you for the equations and the coherent explanation. The bonus of mentioning dispersion,something I'm unfamiliar with, makes this great in my endeavor of further understanding probability. – Michal Frystacky Oct 24 '12 at 17:34

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