Question

EXERCISES ON DISCRETE DISTRIBUTIONS

6. An exam consists of 12 questions that present four possible answers each. A person, without knowledge about the subject of the exam, answers the random exam questions.
a.What is the probability that you get the right answer when answering a question?
b. Find the probability that such person does not answer any questions well
C. Calculate the probability of correcting a question.
d. Obtain the probability that you answer all the questions correctly.
e. Obtain the probability of answering more than half of the questions correctly

Answer 1

Solution:

We are given

n = 12,

p = ¼ = 0.25,

q = 1 – p = 1 – 0.25 = 0.75

P(X=x) = nCx*p^x*q^(n – x)

Part a

a. What is the probability that you get the right answer when answering a question?

We are given that each question has four possible answers.

So, required probability = ¼ = 0.25

b. Find the probability that such person does not answer any questions well

Here, we have to find P(X=0)

P(X=x) = nCx*p^x*q^(n – x)

P(X=0) = 12C0*0.25^0*0.75^(12 – 0)

P(X=0) = 1*1*0.75^12

P(X=0) = 0.031676

C. Calculate the probability of correcting a question.

Here, we have to find P(X=1)

P(X=x) = nCx*p^x*q^(n – x)

P(X=1) = 12C1*0.25^1*0.75^(12 – 1)

P(X=1) = 12*0.25*0.75^11

P(X=1) = 0.126705

d. Obtain the probability that you answer all the questions correctly.

Here, we have to find P(X=12)

P(X=x) = nCx*p^x*q^(n – x)

P(X=12) = 12C12*0.25^12*0.75^(12 – 12)

P(X=12) = 1*0.25^12*1

P(X=12) = 0.0000000596

e. Obtain the probability of answering more than half of the questions correctly

Here, we have to find P(X>6)

P(X>6) = 1 – P(X≤6)

So, by using binomial table for n=12, p = 0.25, we have

P(X≤6) = 0.985747218

P(X>6) = 1 – P(X≤6)

P(X>6) = 1 – 0.985747218

P(X>6) = 0.014252782

Required probability = 0.014252782