In a multiple-choice test, each question has five options. Students will get 5 points for each correct answer; lose 2 point for each incorrect answer; and receive no points for unanswered questions. A student does not know the correct answer for one question. Is it to her advantage or disadvantage to guess an answer? Show your calculations for expected value.
expected value of random choice: 1/5*5 + 4/5*(-2)=1-8/5
expected value of skipping, no answer: 0
so, skip it.
Now what if you could eliminate just one silly wrong answer?
expected value of guessing: 1/4*5-2(3/4)= 1.25-6/4=1.25-1.50=-.25
what if you could eliminate two silly answers....
expected value guessing=1/3*5 -2(2/3)= 5/3-4/3=.3333.
so, if you can't eliminate at least two answers, don't guess on this test. Normally on multiple choice, it is pretty easy to eliminate two answers.
oh okay...so confusing...thank you.
To determine whether it is advantageous or disadvantageous for the student to guess an answer, we need to calculate the expected value of guessing.
The expected value can be calculated by multiplying the probability of each outcome with its corresponding value and then summing them up.
In this case, the student has four potential outcomes for each question they guess.
1. The student guesses correctly:
- Probability of guessing the correct answer = 1/5
- Value for guessing correctly = 5 points
2. The student guesses incorrectly:
- Probability of guessing incorrectly = 4/5
- Value for guessing incorrectly = -2 points
3. The student leaves the question unanswered:
- Probability of leaving the question unanswered = 1 (since it is the only remaining option)
- Value for leaving the question unanswered = 0 points
To calculate the expected value, we multiply the probability by the value for each outcome and sum them up:
Expected value = (Probability of guessing correctly * Value for guessing correctly) + (Probability of guessing incorrectly * Value for guessing incorrectly) + (Probability of leaving unanswered * Value for leaving unanswered)
Expected value = (1/5 * 5) + (4/5 * -2) + (1 * 0)
Simplifying the equation, we get:
Expected value = 1 + (-8/5)
Expected value = -3/5
The expected value of guessing is -3/5, which means on average, the student can expect to lose 3/5 of a point for each question they guess.
Therefore, it is disadvantageous for the student to guess an answer, as they can expect to lose points overall.