Question

an unprepared student takes a four-question, true/false quiz. the student guesses the answers to all four questions, so each answer is equally likely to be correct or wrong. complete parts (a) through (d) below.
(type a whole number.)
are the outcomes in s equally likely?
a. yes. since the student studied for the test, the likelihood the student gets the answer correct is the same as the likelihood the student gets it wrong.
b. no. because the student studied for the test, the likelihood that the student gets more answers correct is higher than the likelihood that the student gets more answers wrong.
c. no. because the student guesses on each of the questions, the likelihood the student gets the answer correct is not the same as the likelihood the student gets it wrong.
d. yes. because the student guesses on each of the questions, the likelihood the student gets the answer correct is the same as the likelihood the student gets it wrong.
(b) write the following event in set notation.
the student gets four answers wrong.
{ }
(use a comma to separate answers as needed.)

Explanation:

Part (a)

The problem is about a true/false quiz where the student guesses. For a true/false question, when guessing, the probability of correct (let's say \( C \)) and wrong (let's say \( W \)) is equal (\( P(C)=P(W) = \frac{1}{2} \)). The student is unprepared and guessing, so each outcome (combination of correct/wrong for 4 questions) should have equal likelihood because each question's correct/wrong is equally likely and independent. Let's analyze the options:

• Option A: Says the student studied, but the problem states the student is unprepared and guessing. So A is wrong.
• Option B: Says the student studied (which is false as per problem) and that correct likelihood is higher, which is wrong.
• Option C: Says correct and wrong likelihoods are not equal, but for guessing true/false, they are equal (\( \frac{1}{2} \) each). So C is wrong.
• Option D: Correctly states that since the student guesses, correct and wrong likelihoods are equal, so outcomes in the sample space \( S \) (all possible answer combinations) are equally likely.

Step 1: Define the sample space elements

For a four - question true/false quiz, let's represent a correct answer as \( C \) and a wrong answer as \( W \). Each question has 2 possibilities, so the sample space has \( 2^4=16 \) elements. But we are interested in the event where the student gets four answers wrong.

Step 2: Represent the event in set notation

If the student gets four answers wrong, then for each of the four questions, the answer is wrong. So the outcome is a sequence of four \( W \)s. So the event (let's call it \( E \)) where the student gets four answers wrong is \( E=\{WWWW\} \)

Answer:

D. Yes. Because the student guesses on each of the questions, the likelihood the student gets the answer correct is the same as the likelihood the student gets it wrong.

Part (b)