Question

2. Assume that random guesses are made for 3 multiple choice questions on a test, so that there are n=3 trials, each with a probability of success given by p=0.2. Find the indicated probability for the number of correct answers. ROUND YOUR ANSWERS TO 3 DECIMAL PLACES a. Find the probability that the number x correct answers will be exactly 3. Q7 b. Find the probability that the number x correct answers will be at least 2. Q8 c. Find the probability that the number x correct answers will be fewer than 2. 09

Answer 1

2.

BIONOMIAL 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
I.
mean = np
where
n = total number of repetitions experiment is executed
p = success probability
mean = 3 * 0.2
= 0.6
II.
variance = npq
where
n = total number of repetitions experiment is executed
p = success probability
q = failure probability
variance = 3 * 0.2 * 0.8
= 0.48
III.
standard deviation = sqrt( variance ) = sqrt(0.48)
=0.6928
a.
probability that the number of correct answers exactly 3
P( X = 3 ) = ( 3 3 ) * ( 0.2^3) * ( 1 - 0.2 )^0
= 0.008
b.
probability that the number of correct answers will be atleast 2
P( X < 2) = P(X=1) + P(X=0)
= ( 3 1 ) * 0.2^1 * ( 1- 0.2 ) ^2 + ( 3 0 ) * 0.2^0 * ( 1- 0.2 ) ^3
= 0.896
P( X > = 2 ) = 1 - P( X < 2) = 0.104
c.
probability that the number of correct answers will be fewer than 2
P( X < 2) = P(X=1) + P(X=0)
= ( 3 1 ) * 0.2^1 * ( 1- 0.2 ) ^2 + ( 3 0 ) * 0.2^0 * ( 1- 0.2 ) ^3
= 0.896