Negative binomial distribution

Negative binomial distribution

Lessons

• Negative Binomial Distribution: P(n)=(n1)C(x1)px(1p)nxP(n)=_{(n-1)}C_{(x-1)}p^x(1-p)^{n-x}
nn: number of trials
xx: number of success in n trials
pp: probability of success in each trial
P(n)P(n): probability of getting the xx success on the nthn^{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?