Suppose there are 40 multiple-choice questions. Each question has 6 possible answers to choose from and only one of the answers is correct. I answer the questions by rolling a fair die for each question and choose the die's facing value as the answer to the questions. Each correct answer is 20 marks and the incorrect answer is -5 mark. I want to find the standard deviation of my total score?
So I got the expected score for $1$ question: $20\times1/6+(-5)\times5/6= -5/6$. And for $40$ questions: $E(Y_i)=40\times-5/6=-100/3$
As each question has the same expected score, I think the standard deviation is 0. But this is wrong, please help me to find standard deviation. Thank you.
Let $X_i$ take the value $1$ if you get question $i$ right, and $0$ otherwise. Then $$X_i\overset{iid}{\sim} Bern(p),$$
where $p=1/6$ in your setup.
Assuming you answer all $n$ questions, your score on the exam is $$S=(20\sum_{i=1}^n X_i)-5(n-\sum_{i=1}^nX_i)=-5n+25\sum_{i=1}^nX_i.$$
Now use the fact that $\sum_{i=1}^n X_i\sim Bin(n,p)\implies \text{Var}(\sum_{i=1}^n X_i)=np(1-p).$