joe takes part in math competitions. a particular contest consists of 25 multiple - choice questions, and…

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 guessed questions with 3 choices

Joe ruled out 2 choices on 4 questions, so there are 3 choices per question. 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 per question is $6\times\frac{1}{3}+0\times\frac{2}{3}= 2$ points. For 4 such questions, the expected points are $4\times2 = 8$ points.

Step3: Calculate expected points from guessed questions with 5 choices

Joe randomly guessed on 9 questions with 5 choices each. 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 per question is $6\times\frac{1}{5}+0\times\frac{4}{5}=1.2$ points. For 9 such questions, the expected points are $9\times1.2 = 10.8$ points.

Step4: Calculate points from blank questions

There are $25-(12 + 4+9)=0$ blank questions, so the points from blank questions are $0\times1.5 = 0$ points.

Step5: Calculate total expected score

The total expected score is the sum of points from sure - answers, guessed questions with 3 choices, guessed questions with 5 choices and blank questions. So the total expected score is $72+8 + 10.8+0=90.8$ points.

Answer:

90.8