I make a multiple choice test with 15 questions and 5 option

I make a multiple choice test with 15 questions and 5 options for each question. For each question, one option is RIDICULOUS, one is CORRECT and three are WRONG BUT NOT RIDICULOUS. I award 5 points for each correct response, and take away 2 points for each ridiculous response. Finally, I take away one point for each wrong non-ridiculous response. Sally is totally unprepared and guesses the answers to each question. What is her expected score?

Solution

Let X₁, X₂,.......,X₁₅ be the scores in the 15 questions.

Consider question 1: probability of choosing correct answer is 1/5, of ridiculous is 1/5 and of wrong is 3/5

E(X₁)=5*1/5+(-2)*1/5+(-1)*3/5 = 0

Also, X_is are all identical in distribution (and independent)

So, E(X₁)=E(X₂)=..................=E(X₁₅)=0

E(X₁+X₂+...............+X₁₅)=E(X₁)+E(X₂)+..................+E(X₁₅)=0

Hence, Sally\'s total expected score is 0.