Assume there are 23 homes in the Quail Creek area and 7 of t

Assume there are 23 homes in the Quail Creek area and 7 of them have a security system. Six homes are selected at random:

What is the probability all six of the selected homes have a security system? (Round your answer to 4 decimal places.)

What is the probability none of the six selected homes has a security system? (Round your answer to 4 decimal places.)

What is the probability at least one of the selected homes has a security system? (Round your answer to 4 decimal places.)

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

Number of HOME that has security system = 7/23 = 0.3043

a)
P( X = 6 ) = ( 6 6 ) * ( 0.3043^6) * ( 1 - 0.3043 )^0
= 0.001

b)

P( X = 0 ) = ( 6 0 ) * ( 0.3043^0) * ( 1 - 0.3043 )^6
= 0.113

c)
P( X > = 1 ) = 1 - P( X < 1) = 0.887
P( X < 1) = P(X=0)   
= ( 6 0 ) * 0.3043^0 * ( 1- 0.3043 ) ^6   
= 0.113