Answer:
a. = 7p - 2
b. = p > 2/7 very disadvantage to guess
a. The expected value of a guess is the sum of the product of the probability of each possible outcome and the associated payoff.
Let p be the probability of answering a question correctly, so the probability of answering a question incorrectly is 1-p.
The expected value of a correct answer is 5p, and the expected value of an incorrect answer is -2(1-p) = -2+2p. Therefore, the expected value of a guess is:
E(g) = 5p - 2(1-p) = 7p - 2
b. To decide if it is to your advantage to guess, we need to compare the expected value of a guess to the expected value of not guessing.
If you don't guess, you don't gain or lose any points, so the expected value of not guessing is 0.
If the expected value of a guess is positive, it is advantageous to guess. If the expected value of a guess is negative, it is disadvantageous to guess.
E(g) > 0
7p - 2 > 0
7p > 2
p > 2/7
Therefore, if you think you have a probability of answering a question correctly greater than 2/7, it is advantageous to guess. If you think your probability of answering the question correctly is less than 2/7, it is better not to guess. If you think your probability of answering the question correctly is exactly 2/7, it doesn't matter whether you guess or not.