Section Binomial Distributions AP Statistics

AP Statistics, Section The Binomial Setting 1. Each observation falls into one of just two categories, which for convenience we call “success” or “failure” 2. There are a fixed number n of observations 3. The n observations are all independent. 4. The probability of success, call it p, is the same for each observation.

AP Statistics, Section The Binomial Setting: Example 1. Each observation falls into one of just two categories, which for convenience we call “success” or “failure”: Basketball player at the free throw. 2. There are a fixed number n of observations: The player is given 5 tries. 3. The n observations are all independent: When the player makes (or misses) it does not change the probability of making the next shot. 4. 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 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 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 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 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 Binomial Coefficients on the Calculator

AP Statistics, Section Binomial Probabilities

AP Statistics, Section Binomial Mean

AP Statistics, Section

AP Statistics, Section

AP Statistics, Section

AP Statistics, Section Exercises , all