Question

A box contains 700 blue and 300 red balls. 200 balls are randomly chosen at the same time, and X is the number of red balls chosen. (a) List the possible values that the random variable X can have. (b) Find the probability mass function of X. (c) Find E[X]. (d) Find Var(X). (e) What is the probability that exactly 50 of the balls chosen are red?

Answer 1

Given that, Box contain 700 Blue and 300 Red balls. Total 1000 balls.And we choose 200 balls randomly at the same time , i.e. without replacement.

Let,

X = number of red balls chosen.

a) Possible values of variable X as

X = {0, 1, 2, ........................., 200}

b) Let, Probability mass function of X is as

p(x=0) = We will have 0 red balls and 200 non red balls =

That is we use Hypergeometric distribution is as

where,

N = is the population size,

K = is the number of success states in the population,

n =  is the number of draws (i.e. quantity drawn in each trial),

k =  is the number of observed successes.

c) Let mean is as

d) Let variance is as

e) Let,

Probability that exactly 50 of the balls chosen are red is 0.0156