Optimal number of answers for a test with wrong-answer penalty Suppose you have to take a test with ten questions, each with four different options (no multiple answers), and a wrong-answer penalty of half a correct answer. Blank questions do not score neither positively nor negatively.
Supposing you have not studied specially hard this time, what's the optimal number of questions to try to answer so the probabilities to pass the exam (having at least five points).

If you answer 5, you need them all right, so your chance (assuming $\frac{1}{4}$ chance of a correct answer) is $(\frac{1}{4})^5=\frac{1}{1024}$.  If you answer 7, you need at least 6 right, with chance $\binom {7}{6}(\frac{1}{4})^6\frac{3}{4}+(\frac{1}{4})^7$, which I make to be about $\frac{1}{745}$.  You can do trying 9, but I think you are in a world of hurt.
