A job applicant has to undergo a test that consists of 6 mul

A job applicant has to undergo a test that consists of 6 multiple choice questions. For each question, 4 possible answers are proposed, of which only one is correct. The candidate knows nothing and decides to guess randomly.

(a) Find the number of different possibilities the candidate has to complete the test.

(b) Find the probability that the candidate answers everything correctly.

(c) Find the probability that the candidate makes only one mistake.

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

a)
P( X = 0 ) = ( 6 0 ) * ( 0.25^0) * ( 1 - 0.25 )^6
= 0.178
P( X = 1 ) = ( 6 1 ) * ( 0.25^1) * ( 1 - 0.25 )^5
= 0.356
P( X = 2 ) = ( 6 2 ) * ( 0.25^2) * ( 1 - 0.25 )^4
= 0.2966
P( X = 3 ) = ( 6 3 ) * ( 0.25^3) * ( 1 - 0.25 )^3
= 0.1318
P( X = 4 ) = ( 6 4 ) * ( 0.25^4) * ( 1 - 0.25 )^2
= 0.033
P( X = 5 ) = ( 6 5 ) * ( 0.25^5) * ( 1 - 0.25 )^1
= 0.0044
P( X = 6 ) = ( 6 6 ) * ( 0.25^6) * ( 1 - 0.25 )^0
= 0.0002

b)
P( X = 6 ) = ( 6 6 ) * ( 0.25^6) * ( 1 - 0.25 )^0
= 0.0002

c)
P( X = 1 ) = ( 6 1 ) * ( 0.25^1) * ( 1 - 0.25 )^5
= 0.356