Question

1. Given a binomial distribution, find the probability of 10 successes out of 15 trials, given p= .85

2. 40% of the population have brown eyes. find the probability that out of 25 people randomly selected at the most 15 have brown eyes.

3. a test consists of 25 multiple choice questions with 5 possible answers, 1 of standard deviation and the probability of getting 8 correct.

Answer 1

1.
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 = 15 * 0.85
= 12.75
II.
variance = npq
where
n = total number of repetitions experiment is executed
p = success probability
q = failure probability
variance = 15 * 0.85 * 0.15
= 1.9125
III.
standard deviation = sqrt( variance ) = sqrt(1.9125)
=1.3829
the probability of 10 successes out of 15 trials
P( X = 10 ) = ( 15 10 ) * ( 0.85^10) * ( 1 - 0.85 )^5
= 0.0449

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 = 25 * 0.4
= 10
II.
variance = npq
where
n = total number of repetitions experiment is executed
p = success probability
q = failure probability
variance = 25 * 0.4 * 0.6
= 6
III.
standard deviation = sqrt( variance ) = sqrt(6)
=2.4495
the probability that out of 25 people randomly selected at the most 15 have brown eyes.
P( X < = 15) = P(X=15) + P(X=14) + P(X=13) + P(X=12) + P(X=11) + P(X=10) + P(X=9) + P(X=8) + P(X=7) + P(X=6) + P(X=5)   
= ( 25 15 ) * 0.4^15 * ( 1- 0.4 ) ^10 + ( 25 14 ) * 0.4^14 * ( 1- 0.4 ) ^11 + ( 25 13 ) * 0.4^13 * ( 1- 0.4 ) ^12 + ( 25 12 ) * 0.4^12 * ( 1- 0.4 ) ^13 + ( 25 11 ) * 0.4^11 * ( 1- 0.4 ) ^14 + ( 25 10 ) * 0.4^10 * ( 1- 0.4 ) ^15 + ( 25 9 ) * 0.4^9 * ( 1- 0.4 ) ^16 + ( 25 8 ) * 0.4^8 * ( 1- 0.4 ) ^17 + ( 25 7 ) * 0.4^7 * ( 1- 0.4 ) ^18 + ( 25 6 ) * 0.4^6 * ( 1- 0.4 ) ^19 + ( 25 5 ) * 0.4^5 * ( 1- 0.4 ) ^20
= 0.9868

3.
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 = 25 * 0.2
= 5
II.
variance = npq
where
n = total number of repetitions experiment is executed
p = success probability
q = failure probability
variance = 25 * 0.2 * 0.8
= 4
III.
standard deviation = sqrt( variance ) = sqrt(4)
=2
the probability of getting 8 correct
P( X = 8 ) = ( 25 8 ) * ( 0.2^8) * ( 1 - 0.2 )^17
= 0.0623