A student is taking a multiplechoice exam in which each ques

A student is taking a multiple-choice exam in which each question has two choices. Assuming that she has no knowledge of the correct answers to any of the question, she has decided on a strategy in which she will place two balls (marked A and B) into a box. She randomly selects one ball for each question and replaces the ball in the box. The making on the ball will determine her answer to the question. There are eight multiple choice questions on the exam. Complete parts(a) through (d) below. What is the probability that she will get eight question correct? (Round to four decimal places as needed) What is the probability that she will get at least seven questions correct? (Round to four decimal places as needed) What is the probability that will get no more than two questions correct? (Round to four decimal places as needed)

Solution

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 = 8 ) = ( 8 8 ) * ( 0.5^8) * ( 1 - 0.5 )^0
= 0.0039

b)
P( X = 7 ) = ( 8 7 ) * ( 0.5^7) * ( 1 - 0.5 )^1
= 0.0313
P( Atleast 7 Right) = P( X = 7 ) + P( X = 8 )
                   = 0.0039 + 0.0313
                   = 0.0352

c)
P( X = 0 ) = ( 8 0 ) * ( 0.5^0) * ( 1 - 0.5 )^8
= 0.0039

d)
P( X < = 2) = P(X=2) + P(X=1) + P(X=0)   
= ( 8 2 ) * 0.5^2 * ( 1- 0.5 ) ^6 + ( 8 1 ) * 0.5^1 * ( 1- 0.5 ) ^7 + ( 8 0 ) * 0.5^0 * ( 1- 0.5 ) ^8
= 0.1445