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**Binomial and geometric Distributions—CH. 8**

Woo—Hoo

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**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.

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**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

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**Binomial Distribution**

A setting where four specific conditions are satisfied is said to be a binomial setting The Binomial Setting 1. 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.

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**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.

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**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).

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Example 1 Blood type is inherited. If both parents carry genes for the O and A blood types, each child has probability 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? ___________________________________________________________________________________________

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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? ____________________________________________________________________________________________________________________________________________________________

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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

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**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.

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**Let’s Try some problems**

Page 516 #’s Are these binomial settings?

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**where n! = ________________________________**

Binomial Coeficient The number of ways of arranging k successes among n observations is given by the binomial coefficient where n! = ________________________________ NOTE: 0 ! = _________

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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?

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**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,

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Try These Page 519 #’s

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Binomial pdf Given a discrete random variable X, the probability distribution function (pdf) assigns a probability to ________________. Calculator HELP: 2nd Vars Binompdf Then enter your data binompdf( n, p, x), Where n is ___________________________, p is __________________________ and x is ________________________________________________.

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**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.

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**Example 5 continued B(_____, _______) distribution…**

We want the probability of making a basket on at most 7 free throws.

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

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**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)

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**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.

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**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?

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**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

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**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

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**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.

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**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.

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**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

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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.

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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

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**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

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**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?

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**Example—Mean and Stand. Dev.**

Let X be the number of tosses until a win. E(X) = _____ = mean p = ___/___ = The variance and standard deviation of X is =

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

Use this:

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

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**Example—Applying Means & Stand. Dev.**

Show ME the MONEY!! In , 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.

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**Practice/Review for Chapter 8**

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Section 6.3 Binomial Distributions. A Gaggle of Girls Let’s use simulation to find the probability that a couple who has three children has all girls.

Section 6.3 Binomial Distributions. A Gaggle of Girls Let’s use simulation to find the probability that a couple who has three children has all girls.

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