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 points from sure - answers

Joe is sure of 12 answers. Each correct answer is worth 6 points. So the points from sure - answers are $12\times6 = 72$ points.

Step2: Calculate expected points from 4 questions with 2 choices ruled out

For each of the 4 questions with 2 choices ruled out, there are $5 - 2=3$ remaining choices. The probability of getting a correct answer is $\frac{1}{3}$, and the probability of getting an incorrect answer is $1-\frac{1}{3}=\frac{2}{3}$. The expected value of points for each question is $6\times\frac{1}{3}+0\times\frac{2}{3}=2$ points. So the total expected points from these 4 questions are $4\times2 = 8$ points.

Step3: Calculate expected points from 9 randomly - guessed questions

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

Step4: Calculate the total expected score

The number of blank questions is $25-(12 + 4+9)=0$. The total expected score is the sum of points from sure - answers, the 4 questions with 2 choices ruled out, and the 9 randomly - guessed questions. So the total expected score is $72+8 + 10.8=90.8$ points.

Answer:

90.8