Chapter 8 8.1 – Binomial Distributions 8.2 - Geometric Distributions AP Statistics Chapter 8 8.1 – Binomial Distributions 8.2 - Geometric Distributions

The Binomial Setting There are 16 games a week in an NFL season (if no teams are on a bye week). They flip a coin at each game to decide who gets the ball first. How many “Tails” will you expect? Will this always happen? If you guess on every question of a 10-question multiple choice quiz, how well do you think you will do? The previous questions dealt with examples of random occurrences that take place in a binomial setting.

Binomial Setting 1. Each observation falls into one of just two categories (often called “success” and “failure”). 2. There is a fixed number, n, of observations. 3. The n observations are all independent. 4. The probability of “success”, usually called p, is the same for each observation.

Binomial Distribution The distribution of the count, X, of successes in the binomial setting… B(n, p) n # of observations p probability of success on any one observation.

Example In 20 rolls of a die, what is the probability of getting exactly 3 fours? Why is this problem difficult to answer based on what you have already learned? Is this a binomial setting? You can’t simply use the multiplication rule, because the fours could be rolled in any 3 of the 20 rolls.

Binomial Coefficient The number of ways of arranging k successes among n observations can be calculated by… Read as “n choose k” In your calculator, n choose k can be found by using the command nCr

Finding Binomial Probabilities X  binomial distribution n  # of observations p  prob of success on each observation

Binomial probabilities on the calculator P(X = k) = binompdf (n, p, k) pdf  probability distribution function  Assigns a probability to each value of a discrete random variable, X. P(X < k) = binomcdf (n, p, k) cdf  cumulative distribution function  for R.V. X, the cdf calculates the sum of the probabilities for 0, 1, 2 … up to k.

Mean and Standard Deviation For a binomial random variable: When n is large, a binomial distribution can be approximated by a Normal distribution. We can use a Normal distribution when. np > 10 and n(1 – p) > 10 If these conditions are satisfied, then a binomial distribution can be approximated by…

The Geometric Setting 1. Each observation falls into one of two categories (“success or “failure”) 2. The observations are independent. 3. The probability of success, p, is the same for all observations. 4. The variable of interest is the number of trials required to obtain the first success. We do not have a fixed number of trials Therefore, the possible values of a geometric random variable are 1, 2, 3, … It is theoretically an infinite set because we may never observe a “success”

Calculating Geometric Probabilities If X has a geometric distribution with probability p of success and (1 – p) of failure on each observation, the possible values of X are 1, 2, 3, … If n is any one of these values, the probability that the first success occurs on the nth trial is:

Calculating Geometric Probabilities The probability that it takes more than n trials to the first success is…

Mean and Standard Deviation If X is a geometric random variable with probability of success p on each trial, then the mean (expected value) of the random variable is: The standard deviation is:

Calculator Functions for Ch 8 Binomial P(X = k)  binompdf(n, p, k) P(X < k)  binomcdf(n, p, k) Geometric P(X < n)  geometcdf(p, n) Normal P(min< X< max) = normalcdf(min, max, μ, σ)