I am trying to generalize and calculate in closed form the best strategy that a gambler should follow to maximize the probability to get a given score in a multiple choice test (i.e. the number of questions to gamble). The problem statement is quite simple and follows Bernoulli process dynamics.
We have a test made by $n$ questions, each question has $r$ alternatives so:
$$ \begin{align} P(\text{answer is correct}) &= \frac{1}{r} \\ P(\text{answer is not correct}) &= 1 - \frac{1}{r} \end{align} $$
the outcomes are the followings:
- 1 point if the answer is correct
- 0 point if the answer is left blank
- $\pi_s < 0$ if the answer is not correct (In most practical cases, $\pi_s$ is equal to a fair value to set the expected value of the score equal to $0$)
And now the problem: what is the number $k \leq n$ of answers to gamble to maximize the probability of getting a final score greater than or equal to a given value $\sigma$? I am interested in a closed form to calculate $k$.
The only way I found to solve this problem is a brute force approach (i.e. calculate the probability for each possible values of $k$ for $\sigma \leq k \leq n$ and then choosing the value of $k$ with best probability). In all my experiments the solution was $k \approx n$, even if a lot of times I got $k \ne n$. Sometimes leaving some question blank is a good idea. I need a way to generalized this in a closed form, if exists...
PS: I deleted original post in mathexchange since I think this is the right community to ask my question.