AP Statistics Chapter 8 Notes

The Binomial Setting If you roll a die 20 times, how many times will you roll a 4? Will you always roll a 4 that many times? If you roll a die 20 times, how many times will you roll a 4? Will you always roll a 4 that many times? The previous questions dealt with an example of a random occurrence that takes place in a binomial setting. The previous questions dealt with an example of a random occurrence that takes place in a binomial setting.

Binomial Setting 1. Each observation falls into one of just two categories (often called “success” and “failure”). 1. Each observation falls into one of just two categories (often called “success” and “failure”). 2. There is a fixed number, n, of observations. 2. There is a fixed number, n, of observations. 3. The n observations are all independent. 3. The n observations are all independent. 4. The probability of “success”, usually called p, is the same for each observation. 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… The distribution of the count, X, of successes in the binomial setting… B(n, p) 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? 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… The number of ways of arranging k successes among n observations can be calculated by… Read as “n choose k” Read as “n choose k” In your calculator, n choose k can be found by using the command n C r In your calculator, n choose k can be found by using the command n C r

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

Binomial probabilities on the calculator P(X = k) = binompdf (n, p, k) P(X = k) = binompdf (n, p, k) pdf  probability distribution function  pdf  probability distribution function  –Assigns a probability to each value of a discrete random variable, X. P(X < k) = binomcdf (n, p, k) P(X < k) = binomcdf (n, p, k) cdf  cumulative distribution function  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: For a binomial random variable: When n is large, a binomial distribution can be approximated by a Normal distribution. When n is large, a binomial distribution can be approximated by a Normal distribution. We can use a Normal distribution when. 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… 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”) 1. Each observation falls into one of two categories (“success or “failure”) 2. The observations are independent. 2. The observations are independent. 3. The probability of success, p, is the same for all observations. 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. 4. The variable of interest is the number of trials required to obtain the first success.

Calculating Geometric Probabilities P(X = n) = (1 – p) n – 1 p P(X = n) = (1 – p) n – 1 p “Probability that the first success occurs on the nth trial” “Probability that the first success occurs on the nth trial” P(X < n)  geometcdf (p, n) P(X < n)  geometcdf (p, n)

Mean and Standard Deviation If X is a geometric random variable with probability of success p on each trial, then If X is a geometric random variable with probability of success p on each trial, then The probability that it takes more than n trials to the first success is… The probability that it takes more than n trials to the first success is… –P(X > n) = (1 – p) n

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