You decide to take your midterm without studying at all. Luckily, all the questi

Question

You decide to take your midterm without studying at all. Luckily, all the questions were of multiple choice. Assume that your chances of guessing the right answer are equal to 25%. If the exam consists of 10 questions and denoting by S the Binomial random variable associated with your score in the exam, then:

a) What is the chance you will earn a score S greater than 8?

b) What is E[S], the expected value of your score?

c) What is the chance that S will be equal to 3?

d) What is the standard deviation of your score?

Explanation / Answer

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

PART A.
P( X < = 8) = P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 10 8 ) * 0.25^8 * ( 1- 0.25 ) ^2 + ( 10 7 ) * 0.25^7 * ( 1- 0.25 ) ^3 + ( 10 6 ) * 0.25^6 * ( 1- 0.25 ) ^4 + ( 10 5 ) * 0.25^5 * ( 1- 0.25 ) ^5 + ( 10 4 ) * 0.25^4 * ( 1- 0.25 ) ^6 + ( 10 3 ) * 0.25^3 * ( 1- 0.25 ) ^7 + ( 10 2 ) * 0.25^2 * ( 1- 0.25 ) ^8 + ( 10 1 ) * 0.25^1 * ( 1- 0.25 ) ^9 + ( 10 0 ) * 0.25^0 * ( 1- 0.25 ) ^10
= 0.99997
P( X > 8) = 1 - P ( X <=8) = 1 -0.99997 = 0.00003

PART B.
I.
mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
mean = 10 * 0.25
= 2.5

PART C.
P( X = 3 ) = ( 10 3 ) * ( 0.25^3) * ( 1 - 0.25 )^7
= 0.25028

PART D.
II.
variance = npq
where
n = total number of repetitions experiment is excueted
p = success probability
q = failure probability
variance = 10 * 0.25 * 0.75
= 1.875
III.
standard deviation = sqrt( variance ) = sqrt(1.875)
=1.36931

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