Question

On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?

Answer 1

Let the number of correct answers be represented by X.

The probability of getting a correct answer is,

p= 1 3

So,

q=1−p =1− 1 3 = 2 3

Then, X has a binomial distribution with n=5 and p= 1 3 .

The probability of x successes P( X=x ) is,

P( X=x )= C n x q n−x p x = C 5 x ( 2 3 ) 5−x ( 1 3 ) x = C 5 x ( 2 3 ) 5−x ( 1 3 ) x

Where x=0,1,…,n.

The probability of guessing more than 4 correct answers is P( X≥4 ),

P( X=4 )+P( X=5 )= C 5 4 ( 2 3 ) ( 1 3 ) 4 + C 5 5 ( 1 3 ) 5 =5⋅( 2 3 )⋅ ( 1 3 ) 4 +1⋅ ( 1 3 ) 5 =5⋅ 2 3 ⋅ 1 81 +1⋅( 1 243 ) = 11 243

Theerefore, the probability of guessing more than 4 correct answer is 11 243 .