Question

l6 pts] Suppose that in the Province of Ontario the written cxam for a drivers license consists ot 4 multiple-choice questions. Each question has 4 possible choices, denoted a,b,cd, only one of which is correct. Fo the sake of simplicity, you may assume throughou ts exerse tha the corret answer is "a for all four quesions, but that a shudent does not know this before taking the exam. Passing requires answering at least 3 queslions correclly. Consider an experiment defined by an uninformed sludent driver guessing randomly on each question, where "guessing randomly means that (i) lhe sludenl answers each queslion independently of the olher queslions and (i) lhal sor each question the student chooses each of the four possible answers with equal probability as the answer. (a) Define the sample space S for the above experiment. (b) Let A denote the event that the student passes the exam. Tahulate the set A. HINT A is นิ sulsel of S; rnake sare il contains all possible scenarios resulting in a passing grade. (c) Tet X he a Be random variable if the student passes (assuming random guessng), and zeo olherwise. Find Pr(X-1). HINT: use your previous fuo ansuers to figure out what is Pr(A)?

Answer 1

(a)

Sample space, S is the set of all possible outcomes of choosing 4 options each for a question.

We have 4 questions. and so, n(S)=4⁴ =256

(b)

A ={(1a,2a,3a,4a), (1a,2a,3a,4b), (1a,2a,3a,4c), (1a,2a,3a,4d), (1a,2a,4a,3b),(1a,2a,4a,3c),(1a,2a,4a,3d),(1a,3a,4a,2b),(1a,3a,4a,2c),(1a,3a,4a,2d),(2a,3a,4a,1b),(2a,3a,4a,1c),(2a,3a,4a,1d)}. So, n(A) =13

(c)

Pr(X=1) =Pr(Student passes) =number of favourable outcomes/total number of outcomes = n(A)/n(S).

n(S) =4⁴ =256

n(A) =n(X=1) =4C₃ *3¹ + 4C₄ *3⁰ =4(3)+1(1) =12+1 =13

Pr(X=1) =Pr(A) =13/256 =0.0508