In an exam there are 10 questions. If you answer correctly to a question, you get $1$ point. If you answer incorrectly to a question, you get $-1$ point, or lose a point. If you don't answer to a question, you get $0$ point. You pass the exam if you get at least $7$ points.
A pupil read the questions and estimated that he can surely answer correctly to $6$ questions, and for the rest questions he estimated that he could answer correctly with independent probabilities $p_1,p_2,p_3,p_4$ with $0\leq p_1,p_2,p_3,p_4\leq 1,p_1p_2p_3p_4=p$. What kind of strategy he must choose to make sure he takes the optimal strategy to pass the exam with respect to numbers $p_1p_2p_3p_4$?
Is that problem solvable? I mean, I figured out the optimal strategy if a student answers correctly with probability $p$ to each of the four questions. This was in Finnish mathematical competition this year. But in this version one has to given that he answers correctly to four question with probability $p$.