# Negative binomial distribution

### Negative binomial distribution

#### Lessons

• Negative Binomial Distribution: $P(n)=_{(n-1)}C_{(x-1)}p^x(1-p)^{n-x}$
$n$: number of trials
$x$: number of success in n trials
$p$: probability of success in each trial
$P(n)$: probability of getting the $x$ success on the $n^{th}$ trial
• Introduction

• Deriving negative binomial distribution
• Formula for negative binomial distribution
• Relation of geometric distribution to the negative binomial distribution

• 1.
Identifying Negative Binomial Distributions
Identify which of the following experiments below are negative binomial distributions?
i.
A fair coin is flipped until head comes up 4 times. What is the probability that the coin will be flipped exactly 6 times?
ii.
Cards are drawn out of a deck until 2 exactly aces are drawn. What is the probability that a total of 10 cards will be drawn?
iii.
An urn contains 3 red balls and 2 black balls. If 2 balls are drawn with replacement what is the probability that 1 of them will be black?
iv.
Roll a die until the first six comes up. What is the probability that this will take 3 rolls?

• 2.
Determining the Negative Binomial Distribution
A fair coin is flipped until head comes up 4 times. What is the probability that the coin will be flipped exactly 6 times?

• 3.
Determining the Cumulative Negative Binomial Distribution
A sculptor is making 3 exhibits for an art gallery. There is a probability of 0.75 that every piece of wood she carves into will be good enough to be part of the exhibit. What is the probability that she uses 4 pieces of wood or less?