Question

on a quiz there are four multiple-choice questions worth 3 points each and two true/false questions worth 1 point each. each multiple-choice question has five possible choices. if a student randomly guesses on each question, what is the expected value of the student’s score on the test?
2.8
3.4
5.4
1.8

Explanation:

Step1: Calculate expected value for multiple - choice questions

For a multiple - choice question with 5 options, the probability of guessing correctly, $p_{mc}=\frac{1}{5} = 0.2$. Each multiple - choice question is worth 3 points. There are 4 multiple - choice questions.
The expected value for one multiple - choice question, $E_{mc}=3\times\frac{1}{5}=\frac{3}{5}=0.6$.
For 4 multiple - choice questions, the total expected value from multiple - choice questions, $E_{mc - total}=4\times0.6 = 2.4$.

Step2: Calculate expected value for true/false questions

For a true/false question, there are 2 options, so the probability of guessing correctly, $p_{tf}=\frac{1}{2}=0.5$. Each true/false question is worth 1 point. There are 2 true/false questions.
The expected value for one true/false question, $E_{tf}=1\times\frac{1}{2}=0.5$.
For 2 true/false questions, the total expected value from true/false questions, $E_{tf - total}=2\times0.5 = 1$.

Step3: Calculate total expected score

The total expected score is the sum of the expected scores from multiple - choice and true/false questions.
$E_{total}=E_{mc - total}+E_{tf - total}=2.4 + 1=3.4$.

Answer:

3.4