Question

A multiple-choice quiz has 150 questions, each with 5 possible answers of which only 1 is correct. What is the probability that sheer guesswork yields from 30 to 35 correct answers for the 90 of the 150 problems about which the student has no knowledge

Answer 1

`\mathbf{P(30 \leq X \leq 35) = 0.00206}`

Explanation:

From the data given;

A multiple-choice quiz has 150 questions

possible answers = 5

correct answer = 1

Let ; The probability of the correct answer = p

Probability of (p) = 1/5

= 0.20

we know that ;

p+q = 1

q = 1 - p

q = 1 - 0.20

q = 0.8

What is the probability that sheer guesswork yields from 30 to 35 correct answers for the 90 of the 150 problems about which the student has no knowledge

We can say that the number of guessed answer should be the sample size n = 90 ,

The probability of getting the correct answer to be between 30 - 35

x = 30,31,32,33,34,35

Using a Binomial Distribution approach to solve this question; we have the formula:

`P(X=x) = nC_X*P^x*q^((n-x))`

Thus;

`P(30 \leq X \leq 35) = P(X=30) +P(X=31)+ P(X=32) +P(X=33) +P(X =34) +P(X =35)`

`P(30 \leq X \leq 35) = 90 C_(30) *(0.20)^(30)*(0.80)^(60)+ 90 C_(31) *(0.20)^(31)*(0.80)^(59)+90 C_(32) *(0.20)^(32)*(0.80)^(58)+90 C_(33) *(0.20)^(33)*(0.80)^(57)+90 C_(34) *(0.20)^(34)*(0.80)^(56)+ 90 C_(35) *(0.20)^(35)*(0.80)^(55)`

`P(30 \leq X \leq 35) = 0.001107+0.000535+0.000247+0.000108+0.000045+0.000018`

`\mathbf{P(30 \leq X \leq 35) = 0.00206}`

Thus; the probability that sheer guesswork yields from 30 to 35 correct answers for the 90 of the 150 problems about which the student has no knowledge is 0.00206