Question

It has been recently estimated that close to40% of Americans have brown eyes. A)Find the probability of having less than 6 brown eyed people in a group of12 Americans. B) Find the probability of having at least 11brown eyed people in a group of 19 Americans. C) Find the probability that in a group of 7 Americans, no one will have brown eyes. D) Find the probability that in a group of 15 Americans, no more than 3 will have brown eyes. E) In a group of 80 Americans, find the probability of having more than 30 brown eyed people.

Answer 1

This is a binomial distribution problem.

But as n increases, it becomes very much tedious to do it by hand..so we do a normal approximation for such problems

For sufficient large enough sample size n, we can use normal approximation for a binomial

If np>5 and n(1 - p) > 5 then we can say that sample is large enough for normal approximation

Here, p = 40% = 0.4 and q = 1 - p = 0.6

So, for sufficient large sample size,  

n > 5/0.4 = 12.5 and n > 5/0.6 = 8.33

So, if sample size n is greater than 12.5 we will use normal approximation in which

Mean = np and standard deviation = sqrt(npq)

Let X be the number of American having brown eyes.