In a region there is a 09 probability chance that a randomly

In a region, there is a 0.9 probability chance that a randomly selected person of the population has brown eyes. Assume 14 people are randomly selected. Complete parts (a) through (d) below.

a. Find the probability that all of the selected people have brown eyes.

The probability that all of the 14 selected people have brown eyes is _____.

(Round to three decimal places as needed.)

b. Find the probability that exactly 13 of the selected people have brown eyes.

The probability that exactly 13 of the selected people have brown eyes is ____.

(Round to three decimal places as needed.)

c. Find the probability that the number of selected people that have brown eyes is 12 or more.

The probability that the number of selected people that have brown eyes is 12 or more is ____.

(Round to three decimal places as needed.)

d. If 14 people are randomly selected, is 12 an unusually high number for those with brown eyes?

A. Yes, because the probability that 12 or more of the selected people have brown eyes is greater than 0.05.

B. No, because the probability that 12 or more of the selected people have brown eyes is greater than 0.05.

C. Yes, because the probability that 12 or more of the selected people have brown eyes is less than 0.05.

D. No, because the probability that 12 or more of the selected people have brown eyes is less than 0.05.

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 = 14 ) = ( 14 14 ) * ( 0.9^14) * ( 1 - 0.9 )^0
= 0.2288
b)
P( X = 13 ) = ( 14 13 ) * ( 0.9^13) * ( 1 - 0.9 )^1
= 0.3559
c)
P( X = 12 ) = ( 14 12 ) * ( 0.9^12) * ( 1 - 0.9 )^2
= 0.257
P(X>=12) = P( X = 12 ) + P( X = 13 ) + P( X = 14 )
       = 0.2288 + 0.3559 + 0.257
       = 0.8416
d)
A. Yes?, because the probability that 12 or more of the selected people have brown eyes is greater than 0.05.