This is all one question on my homework Suppose that you are

This is all one question on my homework.

Suppose that you are in a class of 33 students and it is assumed that approximately 11% of the population is left-handed. (Give your answers correct to three decimal places.)

(a) Compute the probability that exactly five students are left-handed.

(b) Compute the probability that at most four students are left-handed.

(c) Compute the probability that at least six students are left-handed.

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

a)
P( X = 5 ) = ( 33 5 ) * ( 0.11^5) * ( 1 - 0.11 )^28
= 0.1463
b)
P( X < = 4) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 33 4 ) * 0.11^4 * ( 1- 0.11 ) ^29 + ( 33 3 ) * 0.11^3 * ( 1- 0.11 ) ^30 + ( 33 2 ) * 0.11^2 * ( 1- 0.11 ) ^31 + ( 33 1 ) * 0.11^1 * ( 1- 0.11 ) ^32 + ( 33 0 ) * 0.11^0 * ( 1- 0.11 ) ^33
= 0.7052

c)

P( X < 6) = P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 33 5 ) * 0.11^5 * ( 1- 0.11 ) ^28 + ( 33 4 ) * 0.11^4 * ( 1- 0.11 ) ^29 + ( 33 3 ) * 0.11^3 * ( 1- 0.11 ) ^30 + ( 33 2 ) * 0.11^2 * ( 1- 0.11 ) ^31 + ( 33 1 ) * 0.11^1 * ( 1- 0.11 ) ^32 + ( 33 0 ) * 0.11^0 * ( 1- 0.11 ) ^33 +   
= 0.8515
P( X > = 6 ) = 1 - P( X < 6) = 0.1485