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. (a) write out the sample space s, choosing s with equally likely outcomes, if possible. then give the value of n(s) and tell whether the outcomes in s are equally likely. let c represent a correct answer and let w represent a wrong answer. choose the correct answer below. a. s = {ccc, ccw, cwc, wcc, cww, wcw, wwc, www} b. s = {cccc, cccw, ccwc, cwcc, wccc, ccww, cwcw, wccw, wcwc, wwcc, cwwc, cwww, wcww, wwcw, wwwc, wwww} c. s = {cccc,wwww} d. s = {cccc}

Explanation:

We have a four - question true/false quiz, and each question can be either correct (c) or wrong (w). For each of the four questions, there are 2 possible outcomes. By the multiplication principle of counting, the total number of possible sequences of answers (outcomes in the sample space) is \(2\times2\times2\times2 = 2^{4}=16\).

• Option A: The number of elements in set A is 8. Since we have 4 questions, the number of possible outcomes should be \(2^{4} = 16\), so A is incorrect.
• Option B: The set B has 16 elements. Each element is a sequence of 4 characters (either c or w), representing the answers to the 4 questions. This is correct because for each of the 4 questions, we have 2 choices (c or w), and the total number of possible combinations is \(2^{4}=16\).
• Option C: The set C has only 2 elements, which is way less than the total number of possible outcomes (\(2^{4} = 16\)), so C is incorrect.
• Option D: The set D has only 1 element, which is incorrect as there are multiple possible sequences of correct and wrong answers.

Answer:

B. \(S=\{cccc, cccw, c c w c, c w c c, w c c c, ccww, cwcw, wccw, wcwc, wwcc, cwwc, cwww, wcww, wwcw, wwwc, wwww\}\)