Question

joe takes part in math competitions. a particular contest consists of 25 multiple - choice questions, and each question has 5 possible answers. it awards 6 points for each correct answer, 1.5 points for each answer left blank, and 0 points for incorrect answers. joe is sure of 12 of his answers. he ruled out 2 choices before guessing on 4 of the other questions and randomly guessed on the 9 remaining problems. what is his expected score?
69.2
75.6
90.8
97.2

Explanation:

Step1: Calculate score for sure - answers

Joe is sure of 12 answers. Each correct answer awards 6 points. So the score for these is $12\times6 = 72$ points.

Step2: Calculate expected - value for questions with 2 choices ruled out

For the 4 questions where he ruled out 2 choices, there are $5 - 2=3$ choices left. The probability of getting a question correct is $\frac{1}{3}$, and the probability of getting it wrong is $1-\frac{1}{3}=\frac{2}{3}$. The expected value of points per question is $6\times\frac{1}{3}+0\times\frac{2}{3}= 2$ points. So the total expected score for these 4 questions is $4\times2 = 8$ points.

Step3: Calculate expected - value for randomly - guessed questions

For the 9 randomly - guessed questions, the probability of getting a question correct is $\frac{1}{5}$, and the probability of getting it wrong is $1 - \frac{1}{5}=\frac{4}{5}$. The expected value of points per question is $6\times\frac{1}{5}+0\times\frac{4}{5}=\frac{6}{5}=1.2$ points. So the total expected score for these 9 questions is $9\times1.2 = 10.8$ points.

Step4: Calculate score for blank questions

The total number of questions is 25. He has answered $12 + 4+9=25$ questions, so there are $25-(12 + 4 + 9)=0$ blank questions. The score for blank questions is $0\times1.5 = 0$ points.

Step5: Calculate total expected score

The total expected score is the sum of the scores from all types of questions: $72+8 + 10.8=90.8$ points.

Answer:

90.8