Question

An urn contains 5000 balls, of which 100 are yellow and the remaining are purple. We draw 20 balls from the urn and denote the number of yellow balls drawn by X.

(a) What kind of random variable is X if the draws are performed without replacement? Write down the probability distribution function f(x) of X and compute P(X = 5).
(Give an exact expression for the probability that is asked but do not evaluate!)

(b) What kind of random variable is X if the draws are performed with replacement? Write down the probability distribution function f(x) of X and compute P(X ≤ 2). (Give an exact expression for the probability that is asked but do not evaluate!)

(c) Use Poisson distribution to approximate the probability asked in part (b).

Answer 1

(a)

Suppose, random variable X denotes number of yellow balls drawn.

If the draws are performed without replacement, random variable X follows Hypergeometric distribution.

Probability distribution function of X is given by

Required probability is given by

(b)

We can model the given situation using Binomial distribution as follows.

Suppose, random variable X denotes number of yellow balls drawn.

Color of a ball is independent of other balls.

We define getting a yellow ball as success.

So, probability of success is 100/5000=0.02.

Probability distribution function of X is given by

Required probability is given by

(c)

Suppose,

Probability distribution function of X is given by

Required probability is given by