A student takes a test consisting of 100 questions in which the following mark pattern has been set:
- +4 for a correct answer
- -1 for an incorrect answer
- 0 for an unattended question
His marks` range is [-100,400]
Which score will the student get with the most probability if the student has equal probability of either answering it correct or incorrect or leaving the question unanswered?
If there is equal chance of each result, the expectation is $1$ per question and we would expect the maximum of the probability to be about $100$ points. You can get the exact number of ways to get $k$ points by looking at the coefficient of $x^k$ in $$\left(x^{-1}+1+x^4\right)^{100}$$
The relevant section from Alpha is $$ 9383943728600756091550831937222799535986140850 x^{96} + 9441212767563881641238239559852815864411347000 x^{97} + 9478552294706169386338881965717829092395200600 x^{98} + 9495780838072614943530409908336943990394057400 x^{99} + 9492846052391248917058458350660780465837434824 x^{100} + 9469822329629377911072672009202253792816136000 x^{101} + 9426907986531133506321023335178096900208952500 x^{102} + 9364424524015209542017724007185312818138115400 x^{103} + 9282815226893580956739352008183004513923457800 x^{104} + 9182639181075328076178755591937569123430199200 x^{105} $$ and we can see the actual maximum is at $k=99$