Question

Monica claims that she randomly selected answers to ten multiple choice questions (A-E). She correctly answered seven of them. Let X be the number questions answered correctly out of ten

•If she did randomly guess, what’s the probability that any given answer is correct?

•What’s the probability that she guessed 7 questions correctly, by chance alone?

•What’s the probability that she guessed less than 7 questions correctly, by chance alone?

Answer 1

Since the number of options are 5 (A-E), the probability of success, p is 1/5 or 0.2.

Thus, the probability of failure, q is 4/5 or 0.8.

So, the probability that any given answer is correct is 0.2.

We can say that this is a binomial distribution, where n is 10, and p is 0.2.

We must find P(X=7) given that X~N(10,0.2).

nCr * p^r * q^n-r

Where:

n is the number of trial (10 in our case)

r is the number of successes (7 in our case)

p is the probability of success (0.2 in our case)

q is the probability of failure (0.8 in our case)

We may substitute the values in the equation, and the answer we get is 0.000786432.

Thus, the probability that she exactly guessed 7 answers is 0.000786432.

The probability that she guessed than 7 answers ( 0,1,2,3,4,5,6) can be calculated by finding the probabilities for r=0,1,2,3,4,5,6 and adding the probabilities.

A calculation hack is find the probabilities associated with r =7,8,9,10, adding the values up and subtracting it from 1.

P(X=7)= 0.000786432

P(X=8)= 0.000073728

P(X=9)= 0.000004096

P(X=10)= 0.000001

Thus, P(X<7) = 1- P(X=7) + P(X=8) + P(X=9) + P(X=10)

= 1 -  0.000786432 + 0.000073728 + 0.000004096 + 0.000001

= 0.9992924