There are ten multiple choice questions on an examination If

There are ten multiple choice questions on an examination. If there are five choices per question, what is the probability that a student will answer at least five questions correctly just by picking one answer independently at random from the possibilities for each question?

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

P( X < 5) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 10 4 ) * 0.2^4 * ( 1- 0.2 ) ^6 + ( 10 3 ) * 0.2^3 * ( 1- 0.2 ) ^7 + ( 10 2 ) * 0.2^2 * ( 1- 0.2 ) ^8 + ( 10 1 ) * 0.2^1 * ( 1- 0.2 ) ^9 + ( 10 0 ) * 0.2^0 * ( 1- 0.2 ) ^10
= 0.9672
P( Student answers atleast of 5)= P( X > = 5 ) = 1 - P( X < 5) = 0.0328