Question

Bonus Group Project 1: Negative Binomial Distribution

Negative Binomial experiment is based on sequences of Bernoulli trials with probability of success p. Let x+m be the number of trials to achieve m successes, and then x has a negative binomial distribution. In summary, negative binomial distribution has the following properties

• Each trial can result in just two possible outcomes. One is called a success and the other is called a failure.
• The trials are independent
• The probability of success, denoted by p, is the same on every trial.
• The experiment consists of m successes, and x+m repeated trials, and the m^th success occurs at the (x+m)^th trial.

1. Write down the probability distribution P(x=k), consistent with the notation here
2. If you are tossing a regular coin repeatedly, what is the probability that the 3^rd head occurs at the 6^th time you toss it?
3. Anne is selling girl scot cookies in her neighborhood with 20 houses. She has a target to sell 10 boxes. Suppose each house has a probability 0.6 to buy one box of her cookies. What is the probability that she sells the last box at the 15^th house? What is the probability that she exhausts all 20 houses?

Answer 1

Given , X+m is the number of trials to achieve mth success (X be the number of failures before mth success) , with probability of success is p

X follow Negative Binomial distribution

The probability distribution of X is given by

Tossing a coin

Let X be the number of failures before the third head ( or X+3 be the number of trials), m=3 , p= 0.5 (probability of head)

To find P(X=3) =?

Probability that 3rd head occurs at 6th toss = 0.15625

Selling cookies

Let X+ 10 be the number of house she goes to sells 10 boxes

X follow negative Binomial distribution with m=10 , p =0.6

Probability that she sells the last box at 15th house is 0.12396

Probability that she exhausts 20 houses = 0.05857