BINOMIAL AND GEOMETRIC DISTRIBUTIONS—CH. 8

Woo—Hoo

The Binomial Distribution Two different classes of distribution; binomial and

geometric Success/failure observations, such as…

a coin toss to see which of the two football teams gets the choice of kicking

basketball player shoots a free throw {makes the shot, misses the shot}.

A young couple prepares for their first child…a boy or a girl.

quality control inspector selects a widget coming off the assembly line; he is interested in whether or not the widget meets production requirements.

The Binomial Distribution

We will use what we have learned about probability and random variables to complete the necessary foundation toward studying inference.

Remember: an independent event is where one outcome has no influence on another outcome

Binomial Distribution

A setting where four specific conditions are satisfied is said to be a binomial setting

The Binomial Setting1. Each observation falls into one of just two

categories, which for convenience we call “success” or “failure”.

2. There is a fixed number of observations (n).3. The observations are independent. 4. The probability of success (p) is the same for each

observation.

Binomial Distribution

You need to know and understand the four properties of a binomial setting for later insight

If it’s a binomial setting, the random variable X = number of successes is called a binomial random variable.

AND the probability distribution of X is called a binomial distribution.

Binomial Distribution

Binomial Distribution The distribution of the count of X of successes in the

binomial setting is the binomial distribution with parameters n and p.

The parameter n is the number of observations, and p is the probability of a success on any one observation.

The possible values of X are the whole numbers from 0 to n. As an abbreviation, we say that X is B (n, p).

Example 1

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 independent of each other. A couple will have four children, and are interested in the number of children that inherit blood type O.

Is it reasonable to assume a binomial distribution in this situation?

______________________________________________________________________________________________________________________________________________________________________________________

Example 2

Deal 10 cards from a shuffled deck and count the number X of red cards. There are 10 observations, and each gives either a red or black card. A “success” is a red card. A card is drawn and not replaced before the next card is drawn.

Is it reasonable to assume a binomial distribution in this situation?

____________________________________________________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________________________________________________

Read example 3 on page 515 In this example and later in the chapter

we see that although not strictly independent and therefore not a binomial setting… As long as the sample size is small compared

to the population, this is an option

Sampling Distribution of a Count: This idea of “close enough” is common in statistical reasoning, but not in typical math classes

Sampling Distribution of a Count Choose an SRS of size n from a

population with proportion p of successes. ____________________ ________________________________________________________________________________, the count X of successes in the sample has approximately the binomial distribution with parameters n and p.

Let’s Try some problems

Page 516 #’s 8.1-8.6 Are these binomial settings?

Binomial Coeficient

The number of ways of arranging k successes among n observations is given by the binomial coefficient

where n! = ________________________________

NOTE: 0 ! = _________

)!(!

!

knk

n

k

n

Example 4

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 independent of each other.

A couple will have four children. What is the probability that exactly 2 of them will have type O blood?

Binomial Probability (needed for Ex.4)

If X has the binomial distribution with n observations and probability p of success on each observation, the possible values of X are 0, 1, 2, …, n. If k is any one of these values,

knk ppk

nkXP

)1()(

Try These

Page 519 #’s 8.7-8.11

Binomial pdf

Given a discrete random variable X, the probability distribution function (pdf) assigns a probability to ________________.

Calculator HELP:2nd VarsBinompdf

Then enter your data binompdf( n, p, x), Where n is ___________________________, p is __________________________ and x is ________________________________________________.

Example 5—calculating binomial probabilities

Corinne is a basketball player who makes 75% of her free throws in a season. In a key game, she shoots 12 and makes 7 free throws. Fans think she was nervous. Is it unusual for her to perform this poorly?

To answer this question, assume that the free throws are independent with a probability of 0.75 for each success. The number X of baskets (successes) in 12 attempts has the B(______, ______) distribution.

Example 5 continued

B(_____, _______) distribution… We want the probability of making a

basket on at most 7 free throws.

1576.0)7(

1032.00401.00115.00024.00004.00000.00000.00000.0)7(

___)(...___)(___)()7(

XP

XP

XPXPXPXP

Binomial cdf

Given a random variable X, the cumulative distribution function (cdf) of X calculates the sum of the probabilities for 0, 1, 2, …, up to the value of X.

Calculator HELP:2nd VarsBinomcdf

Then enter your data binomcdf( n, p, x), Where n is_______________________ p is ______________________________ and x is __________________PLUS _____________________________.

Example 5—calculating binomial probabilities USING a Calculator On calculator B(12, 0.75)…where we

stop at 7 (including 7) 2nd

Vars Scroll down to find binomcdf (12, 0.75, 7)

Mean and Standard Deviation of a Binomial Random Variable

If a count X has the binomial distribution with number of observations n and probability of success p, the mean and standard deviation of X are

* These short formulas _____________BINOMIAL distributions! They ____________ be used for other

discrete random variables.

np

)1( pnp

Example 4 Revisited

BLOOD TYPES What is the expected number of children

the couple will have with type O blood? What is the standard deviation of the

number of children the couple will have with type O blood?

Normal Approximation for Binomial Distributions When n is large, binomial distributions

can be considered approximately normal. A count X has the binomial distribution

with n trials and success probability p. When n is __________…

As a rule of thumb we will use the Normal approximation when n and p satisfy

))1(,( pnpnpN

10)1(

10

pn

np

8.2 The Geometric Distribution

The binomial variable X counts the __________________ in that ____________________of trials. If there are n trials, then the possible values of X are 0, 1, 2, …, n.

Geometric Random Variable If our goal is to obtain ________ success, a random

variable X can be defined that counts the number of trials needed to obtain the __________ success. A random variable that satisfies this description is a geometric random variable.

p.533 example 8.14

Examples of an Infinite Situation Flip a coin until you get a head. Roll a die until you get a 3. In basketball, attempt a three-point shot

until you make a basket.

The four Characteristics

The Geometric Setting 1. Each observation falls into one of just

two categories, which for convenience we call “success” or “failure.”

2. The probability of a success (p) is the same for each observation.

3. The observations are all independent. 4. The variable of interest is the number

of trials required to obtain the first success.

Geometric Probability Formula

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

The probability that it takes more than n trials is

ppnXP n 1)1()(

npnXP )1()(

Example 1—DRAW AN ACE Suppose you repeatedly draw cards with

replacement from a deck of 52 cards until you draw an ace.

Is it reasonable to assume a geometric distribution in this situation?

  Calculate the probability of drawing an

ace on the first draw.

Example 1—DRAW AN ACE

Calculate the probability of drawing an ace on the second draw.

Construct a probability distribution table for DRAW AN ACE.

 

Construct a probability histogram for DRAW AN ACE.

X: 1 2 3 4 5 6 …

P(X): p (1-p)p

(1-p)2p

(1-p)3p (1-p)4p

(1-p)5p

Calc. trick to find the probability distribution info on PAGE543

Mean and Standard Deviation of a Geometric Random Variable

The mean, or expected value of the geometric random variable is the ______________________________ required to get the ____________.

The standard deviation of a geometric random variable is

px

1

2

)1(

p

px

Example—Mean and Stand. Dev.

ARCADE GAME—Glenn likes the game at the state fair where you toss a coin in to a saucer. You win if the coin comes to rest in the saucer without sliding off. Glenn has played this game many times and has determined that on average he wins 1 out of every 12 times he plays. He believes that his chances of winning are the same for each toss. He has no reason to think that he tosses are not independent.

What is the expected number of tosses Glenn will have to toss in order to win a game?

What is the standard deviation of the number of tosses?

Example—Mean and Stand. Dev. Let X be the number of tosses until a

win. E(X) = _____ = mean p = ___/___ = 0.0833

The variance and standard deviation of X is

---------- =

x

x p

p2

2 1

What if it takes more than n trials to see the first success?

Use this:

npnXP )1()(

)( nXP

Technology Tip

This equation is not quite as easy to calculate, so using the following setup in the calculator is helpful.

Calculator:

),()( npgeometcdfnXP

)1(1)( pnXP n

Example—Applying Means & Stand. Dev.Show ME the MONEY!! In 1986-1987, cheerios cereal boxes displays

a dollar bill on the front of the box and a cartoon character who said, “Free $1 bill in every 20th box.” Use the simulation chart on page 549 to determine the number of boxes of Cheerios you would expect to buy in order to get one of the “free” dollar bills.

Practice/Review for Chapter 8 P.529 #8.21 P.534 #8.28 P.537 #8.34 P.543 # 8.41 P.550 #8.48 & 8.49 P.552 #8.53 & 8.54 P.558 #8.64-8.66