Question

An urn contains 10 red balls and 5 green balls. Balls are randomly selected, one at a time, with replacement, until a red one is obtained. What is the probability that exactly k draws are needed? What is the probability that at least k draws are needed? Define a random variable associated with this experiment. Determine its probability mass function and cumulative distribution function, sketch their graphs. Find the expectation, variance and standard deviation of X.

Answer 1

An urn contains 10 red balls and 5 green balls.

Therefore, total number of balls = 15

Balls are randomly selected, one at a time, with replacement.

Therefore, the probability of selecting a red ball = p = 10/15 = 2/3

Let X be the random variable denoting the number of draws required to obtain a red ball from the urn.

Therefore, X follows the geometric distribution with parameter p.

The probability that exactly k draws are needed is given by,

The probability that at least k draws are needed is given by,