please answer letter d moslty and the others that i got part

please answer letter d moslty and the others that i got partialy wrong please a, b, and c

When looking at a person\'s eye color, it turns out that 1% of people in the world have green eyes. Consider a group of 15 people and we want to examine how many of them have green eye colors. What is the random variable Y in the case? What is the distribution of Y? Find the probability that NONE of them has greens eyes. Find the probability that at least one of them has green eyes. What are the expected value and standard deviation for this random variable?

Solution

a).

n=15

P=0.01

Binomial distribution is used.

b).

P(X=x) = (_nC_x) p^x (1-p)^n-x   

P(X=0) = (₁₅C₀) 0.01⁰ (1-0.01)^15-0   

P(X = 0) = 0.8601

c).

P( atleast one of them green eyes) = 1-P( X=0)

= 1-0.8601

= 0.1399

d).

Expectation = np = 0.15

Variance = np(1 - p) = 0.1485

Standard deviation = 0.3854