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
57.2

Explanation:

Step1: Calculate score from sure answers

Joe is sure of 12 answers, each correct gives 6 points. So score: \(12\times6 = 72\)

Step2: Analyze 4 guessed questions (2 choices ruled out)

Each has 5 - 2 = 3 choices, so probability of correct: \(\frac{1}{3}\), incorrect: \(\frac{2}{3}\). Expected points per question: \(6\times\frac{1}{3}+0\times\frac{2}{3}=2\). For 4 questions: \(4\times2 = 8\)

Step3: Analyze 9 randomly guessed questions

Each has 5 choices, probability of correct: \(\frac{1}{5}\), incorrect: \(\frac{4}{5}\). Expected points per question: \(6\times\frac{1}{5}+0\times\frac{4}{5}=1.2\). For 9 questions: \(9\times1.2 = 10.8\)

Step4: Calculate total expected score

Sum all parts: \(72 + 8+10.8=90.8\)? Wait, no, wait: Wait, total questions: 25. Sure:12, 4 guessed (ruled 2), 9 random. Wait, wait, maybe I miscounted. Wait, 12 + 4 + 9 = 25. But wait, the blank? Wait, no, the problem says: "6 points for each correct answer, 1.5 points for each answer left blank, and 0 points for incorrect answers." Wait, did Joe leave any blank? Wait, the problem says: "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." So all questions are either sure (correct), guessed (4 with 3 choices, 9 with 5 choices). So no blanks. So recalculate:

Sure: 12 correct: \(12\times6 = 72\)

4 questions with 3 choices: expected correct per question: \(\frac{1}{3}\), so expected points: \(4\times(6\times\frac{1}{3}+0\times\frac{2}{3})=4\times2 = 8\)

9 questions with 5 choices: expected correct per question: \(\frac{1}{5}\), so expected points: \(9\times(6\times\frac{1}{5}+0\times\frac{4}{5})=9\times1.2 = 10.8\)

Wait, but 72 + 8 + 10.8 = 90.8? But the options have 90.8 as an option. Wait, but let me check again. Wait, maybe I made a mistake. Wait, no, wait: Wait, the 4 questions: he ruled out 2 choices, so 3 left. So probability of correct is 1/3. So expected value per question: 6(1/3) + 0(2/3) = 2. So 4*2=8.

9 questions: 5 choices, so 1/5 chance correct. Expected per question: 6(1/5)=1.2. 91.2=10.8.

Sure: 12*6=72.

Total: 72 + 8 + 10.8 = 90.8. Wait, but the options include 90.8. Wait, but maybe I messed up. Wait, no, let's check again.

Wait, the problem: 25 questions. Sure:12, 4 guessed (ruled 2), 9 random. So all answered, no blanks. So:

Sure: 12 correct: 12*6=72.

4 questions: each has 3 options, so P(correct)=1/3, P(incorrect)=2/3. Expected points per question: 6(1/3) + 0(2/3)=2. So 4*2=8.

9 questions: 5 options, P(correct)=1/5, P(incorrect)=4/5. Expected points per question: 6(1/5)=1.2. 91.2=10.8.

Total expected: 72 + 8 + 10.8 = 90.8. So that's the answer.

Answer:

90.8