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Section 8.1.1 Binomial Distributions

Section 8.1.1 Binomial Distributions. AP Statistics www.toddfadoir.com/apstats. The Binomial Setting. Each observation falls into one of just two categories, which for convenience we call “success” or “failure” There are a fixed number n of observations

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Section 8.1.1 Binomial Distributions

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  1. Section 8.1.1Binomial Distributions AP Statistics www.toddfadoir.com/apstats

  2. The Binomial Setting • Each observation falls into one of just two categories, which for convenience we call “success” or “failure” • There are a fixed number n of observations • The n observations are all independent. • The probability of success, call it p, is the same for each observation. AP Statistics, Section 8.1.1

  3. The Binomial Setting: Example • Each observation falls into one of just two categories, which for convenience we call “success” or “failure”: Basketball player at the free throw. • There are a fixed number n of observations: The player is given 5 tries. • The n observations are all independent: When the player makes (or misses) it does not change the probability of making the next shot. • The probability of success, call it p, is the same for each observation: The player has an 85% chance of making the shot; p=.85 AP Statistics, Section 8.1.1

  4. Shorthand • Normal distributions can be described using the N(µ,σ) notation; for example, N(65.5,2.5) is a normal distribution with mean 65.5 and standard deviation 2.5. • Binomial distributions can be described using the B(n,p) notation; for example, B(5, .85) describes a binomial distribution with 5 trials and .85 probability of success for each trial. AP Statistics, Section 8.1.1

  5. Example • Blood type is inherited. If both parents carry genes for the O and A blood types, each child has probability 0.25 of getting two O genes and so of having blood type O. Different children inherit independently of each other. The number of O blood types among 5 children of these parents is the count X off successes in 5 independent observations. • How would you describe this with “B” notation? • X=B(5,.25) AP Statistics, Section 8.1.1

  6. Example • Deal 10 cards from a shuffled deck and count the number X of red cards. These 10 observations, and each gives either a red or a black card. A “success” is a red card. • How would you describe this using “B” notation? • This is not a Binomial distribution because once you pull one card out, the probabilities change. AP Statistics, Section 8.1.1

  7. Binomial Coefficient • Sometimes referred to as “n choose k” • For example: “I have 10 students in a class. I need to choose 2 of them.” • In these examples, order is not important. AP Statistics, Section 8.1.1

  8. Binomial Coefficients on the Calculator AP Statistics, Section 8.1.1

  9. Binomial Probabilities AP Statistics, Section 8.1.1

  10. Binomial Mean AP Statistics, Section 8.1.1

  11. AP Statistics, Section 8.1.1

  12. AP Statistics, Section 8.1.1

  13. AP Statistics, Section 8.1.1

  14. Exercises • 8.1-8.7, all AP Statistics, Section 8.1.1

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