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).

Let's say that you have a probability $p$ of getting the correct answer to each question. Then if you answer $n$ of the questions, the number that you answer correctly is distributed as a binomial random variable $X\sim B(n,p)$.
Your score on the exam, $Y$, is related to this. You score 1 for each correct answer and lose 1/2 for each incorrect answer, which means that
$$Y = X - \tfrac{1}{2}(n-X) = \tfrac{1}{2}(3X-n)$$
You are asking for the probability that $Y\geq 5$, which reduces to
$$\begin{align}
P(Y\geq 5)
& = P\left(\frac{3X-n}{2} \geq 5\right) \\
& = P\left(X \geq \frac{10+n}{3}\right)
\end{align}$$
This probability can be easily calculated from the binomial density, or (cheating) using the following R code:
> f <- function(n,p) {pbinom((10+n)/3 - 0.001, n, p, lower.tail=FALSE)}
> plot(0:10, f(0:10, 0.25), type='l')

which generates this chart, from which you can see that you should answer eight questions:
[image upload not working at the moment, I'll come back and edit the image in later]
