Probability of guessing True-False question using different test-taking strategies. 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
 A: 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.
A: 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. 
