On a certain statistics exam, there are 13 multiple choice questions. Each quest

Question

On a certain statistics exam, there are 13 multiple choice questions. Each question has 4 possible answers to choose from (and only 1 correct answer). Suppose you didn’t study at all and are planning on guessing the correct answer for each question.

Therefore, the probability of guessing a question correctly is . (Round to two decimals)

Let X = number of questions the student guesses correctly

(Round all answers to four decimals)

a)      What’s the probability you guess at least one question correctly?

b)      What’s the probability you guess strictly less than half of the questions correctly?

c)      What’s the expected number of questions you guess correctly?

d)     What’s the probability of guessing between 1 and 5 questions (inclusive) correctly?

Explanation / Answer

Let X denote the number of correct guesses.

Here success probability p = 0.25

(a)

P(X>=1) = 1 - P(X=0)

P(X=0) =  13C0.p0.(1-p)13-0 = 1*0.250*0.7513 = 0.0237

So,

P(X>=1) = 1 - 0.0237 = 0.9763

(b)

Half of the questions means 6.5 or about 6. Less than half means 5.

P(X<=5) =P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)

P(X=1) =  13C1.p1.(1-p)13-1 = 13*0.251*0.7512 = 0.103

P(X=2) =  13C2.p2.(1-p)13-2 = 78*0.252*0.7511 = 0.206

P(X=3) =  13C3.p3.(1-p)13-3 = 286*0.253*0.7510 = 0.251

P(X=4) =  13C4.p4.(1-p)13-4 = 715*0.254*0.759 = 0.209

P(X=5) =  13C5.p5.(1-p)13-5 = 1287*0.255*0.758 = 0.072

So,

P(X<=5) = 0.0237 + 0.103 + 0.206 + 0.251 + 0.209 + 0.072 = 0.865

(c)

E(X) = n*p = 13*0.25 = 3.25 correct questions

(d)

P(1<=X<=5) = P(X<=5) - P(X=0) = 0.865 - 0.0237 = 0.8413

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