AP StatisticsAP Statistics

Chapter 8 NotesChapter 8 Notes

The Binomial SettingThe Binomial Setting

If you roll a die 20 times, how many If you roll a die 20 times, how many times will you roll a 4? Will you times will you roll a 4? Will you always roll a 4 that many times?always roll a 4 that many times?

The previous questions dealt with an The previous questions dealt with an example of a random occurrence example of a random occurrence that takes place in a that takes place in a binomial binomial setting.setting.

Binomial SettingBinomial Setting

1. Each observation falls into one of just 1. Each observation falls into one of just two categories (often called “success” and two categories (often called “success” and “failure”).“failure”).

2. There is a fixed number, n, of 2. There is a fixed number, n, of observations.observations.

3. The n observations are all independent.3. The n observations are all independent. 4. The probability of “success”, usually 4. The probability of “success”, usually

called p, is the same for each observation.called p, is the same for each observation.

Binomial DistributionBinomial Distribution

The distribution of the count, X, of The distribution of the count, X, of successes in the binomial setting…successes in the binomial setting…

B(n, p)B(n, p)– nn # of observations # of observations– pp probability of success on any one probability of success on any one

observation.observation.

ExampleExample

In 20 rolls of a die, what is the In 20 rolls of a die, what is the probability of getting exactly 3 fours?probability of getting exactly 3 fours?– Why is this problem difficult to answer Why is this problem difficult to answer

based on what you have already based on what you have already learned?learned?

– Is this a binomial setting?Is this a binomial setting?– You can’t simply use the multiplication You can’t simply use the multiplication

rule, because the fours could be rolled in rule, because the fours could be rolled in any 3 of the 20 rolls.any 3 of the 20 rolls.

Binomial CoefficientBinomial Coefficient

The number of ways of arranging k successes The number of ways of arranging k successes among n observations can be calculated by…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 In your calculator, n choose k can be found by

using the command using the command nnCCrr

Finding Binomial Finding Binomial ProbabilitiesProbabilities

X X binomial distribution binomial distribution n n # of observations # of observations p p prob of success on each prob of success on each

observationobservation

Binomial probabilities on the Binomial probabilities on the calculatorcalculator

P(X = k) = binompdf (n, p, k)P(X = k) = binompdf (n, p, k) pdf pdf probability distribution function probability distribution function

– Assigns a probability to each value of a Assigns a probability to each value of a discrete random variable, X.discrete random variable, X.

P(X P(X << k) = binomcdf (n, p, k) k) = binomcdf (n, p, k) cdf cdf cumulative distribution function cumulative distribution function

– for R.V. X, the cdf calculates the sum of the for R.V. X, the cdf calculates the sum of the

probabilities for 0, 1, 2 … up to k.probabilities for 0, 1, 2 … up to k.

Mean and Standard Mean and Standard DeviationDeviation

For a binomial random For a binomial random variable:variable:

When n is large, a When n is large, a binomial distribution can binomial distribution can be approximated by a be approximated by a Normal distribution.Normal distribution.

We can use a Normal We can use a Normal distribution when.distribution when.– np np >> 10 and n(1 – p) 10 and n(1 – p) >> 10 10

If these conditions are If these conditions are satisfied, then a satisfied, then a binomial distribution can binomial distribution can be approximated by…be approximated by…

The Geometric SettingThe Geometric Setting

1. Each observation falls into one of 1. Each observation falls into one of two categories (“success or “failure”)two categories (“success or “failure”)

2. The observations are independent.2. The observations are independent. 3. The probability of success, p, is 3. The probability of success, p, is

the same for all observations.the same for all observations. 4. The variable of interest is the 4. The variable of interest is the

number of trials required to obtain number of trials required to obtain the first success.the first success.

Calculating Geometric Calculating Geometric ProbabilitiesProbabilities

P(X = n) = (1 – p)P(X = n) = (1 – p)n – 1n – 1pp

““Probability that the first success Probability that the first success occurs on the nth trial”occurs on the nth trial”

P(X P(X << n) n) geometcdf (p, n) geometcdf (p, n)

Mean and Standard Mean and Standard DeviationDeviation

If X is a geometric random variable with If X is a geometric random variable with probability of success p on each trial, probability of success p on each trial, thenthen

The probability that it takes more than The probability that it takes more than n trials to the first success is…n trials to the first success is…– P(X > n) = (1 – p)P(X > n) = (1 – p)nn

Calculator Functions for Ch Calculator Functions for Ch 88

BinomialBinomial– P(X = k) P(X = k) binompdf(n, p, k) binompdf(n, p, k)– P(X P(X << k) k) binomcdf(n, p, k) binomcdf(n, p, k)– Simulation Simulation randbin(n, p) randbin(n, p)

GeometricGeometric– P(X P(X << n) n) geometcdf(p, n) geometcdf(p, n)

NormalNormal– P(min< X< max) = normalcdf(min, max, P(min< X< max) = normalcdf(min, max,

μμ, , σσ))