In a math contest problem appeared which I have trouble solving . It goes as under - Consider an examination of $N$ questions - fully multiple choice questions . There are $c$ choices for each question. And marking for each question scheme as follows -
If the candidate chose the answer correctly then he get $+M$ and for every false answer he gives he receives $-m$. (Where $M>m)$. Given that he guesses all the questions , calculate the probability of him scoring a positive total .
Let $x$ denote the positive number of guesses.
For positive score: $Mx-m(N-x) >0 \implies x>\frac{mN}{M+m}$
Probability that his guess is correct is $\frac{1}{c}$
Then, the required probability for scoring positive marks is $(\frac{1}{c})^{\lfloor\frac{mN}{M+m}\rfloor}$