Suppose there are 4 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 $5$ marks and incorrect answer is $-1$ mark. I want to find the expected score I will get.
So, I could let $X\sim Binomial(4,\frac{1}{6})$.
Then, $E(X)=4(\frac{1}{6})=\frac{2}{3}$. This is the average number of questions that I will be correct.
Since each correct answer is $5$ points and wrong answer is $-1$ point, $\frac{2}{3} \cdot 5 + (6-\frac{2}{3}) \cdot (-1) = -3.333$. So I say that my expected score is $-3.333$.
But on a second thought, if I let each question be $Y_i$. So there are four questions, so $Y_1, ..., Y_4$. And the expected score for each question is: $E(Y_i)= \frac{1}{6} \cdot 5 + \frac{5}{6} \cdot (-1) = 0$
Then for four questions, $4 \cdot E(Y_i) = 4 \cdot 0 = 0$. In this case, then my expected score is $0$.
Now, I am confuse. How should I think about the problem to get the expected score?