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# Bonus Group Project 1: Negative Binomial Distribution Negative Binomial experiment is based on sequences of Bernoulli...

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 mth 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 3rd head occurs at the 6th 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 15th house? What is the probability that she exhausts all 20 houses?

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

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

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

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