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**Another Random Variable**

Suppose that each of three randomly selected customers purchasing a hot tub at a certain store chooses either an electric (E) or a gas (G) model. Assume that these customers make their choices independently of one another and that 40% of all customers select an electric model. The number among the three customers who purchase an electric hot tub is a random variable. What is the probability distribution?

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**Chapter 17 Binomial and Geometric Distributions**

Binomial vs. Geometric Chapter 17 Binomial and Geometric Distributions

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**Binomial vs. Geometric The Binomial Setting The Geometric Setting**

Each observation falls into one of two categories. 1. Each observation falls into one of two categories. The probability of success is the same for each observation. 2. The probability of success is the same for each observation. The observations are all independent. The observations are all independent. There is a fixed number n of observations. 4. The variable of interest is the number of trials required to obtain the 1st success.

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**Are Random Variables and Binomial Distributions Linked?**

X = number of people who purchase electric hot tub X P(X) .216 .432 .288 .064 GGG (.6)(.6)(.6) EEG GEE EGE (.4)(.4)(.6) (.6)(.4)(.4) (.4)(.6)(.4) EGG GEG GGE (.4)(.6)(.6) (.6)(.4)(.6) (.6)(.6)(.4) EEE (.4)(.4)(.4)

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Combinations Formula: Practice:

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**M&Ms According to Mars a few years ago, 25% of M&Ms are orange.**

You pull three M&Ms out of a bag.

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**Developing the Formula**

Outcomes Probability Rewritten OcOcOc OOcOc OcOOc OcOcO OOOc OOcO OcOO OOO

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**Developing the Formula**

n = # of observations p = probability of success k = given value of variable Rewritten

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**Working with probability distributions**

State the distribution to be used Define the variable State important numbers Binomial: n & p Geometric: p Normal: m & s

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**X = # of people use express**

Twenty-five percent of the customers entering a grocery store between 5 p.m. and 7 p.m. use an express checkout. Consider five randomly selected customers, and let X denote the number among the five who use the express checkout. binomial n = p = .25 X = # of people use express

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**What is the probability that two used express checkout?**

binomial n = p = .25 X = # of people use express

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**What is the probability that at least four used express checkout?**

binomial n = p = .25 X = # of people use express

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**X = # of people who believe…**

“Do you believe your children will have a higher standard of living than you have?” This question was asked to a national sample of American adults with children in a Time/CNN poll (1/29,96). Assume that the true percentage of all American adults who believe their children will have a higher standard of living is .60. Let X represent the number who believe their children will have a higher standard of living from a random sample of 8 American adults. binomial n = p = .60 X = # of people who believe…

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**Interpret P(X = 3) and find the numerical answer.**

binomial n = p = .60 X = # of people who believe The probability that 3 of the people from the random sample of 8 believe their children will have a higher standard of living.

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**X = # of people who believe**

Find the probability that none of the parents believe their children will have a higher standard. binomial n = p = .60 X = # of people who believe

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**Binomial vs. Geometric The Binomial Setting The Geometric Setting**

Each observation falls into one of two categories. 1. Each observation falls into one of two categories. The probability of success is the same for each observation. 2. The probability of success is the same for each observation. The observations are all independent. The observations are all independent. There is a fixed number n of observations. 4. The variable of interest is the number of trials required to obtain the 1st success.

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**Developing the Geometric Formula**

X Probability

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**X = # of disc drives till defective**

Suppose we have data that suggest that 3% of a company’s hard disc drives are defective. You have been asked to determine the probability that the first defective hard drive is the fifth unit tested. geometric p = .03 X = # of disc drives till defective

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**X = # of free throws till miss**

A basketball player makes 80% of her free throws. We put her on the free throw line and ask her to shoot free throws until she misses one. Let X = the number of free throws the player takes until she misses. geometric p = .20 X = # of free throws till miss

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**What is the probability that she will make 5 shots before she misses?**

geometric p = .20 X = # of free throws till miss What is the probability that she will miss 5 shots before she makes one? geometric p = .80 Y = # of free throws till make

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**X = # of free throws till miss**

What is the probability that she will make at most 5 shots before she misses? geometric p = .20 X = # of free throws till miss

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**What is the probability that she will make at least 8 shots before she misses?**

geometric p = .20 X = # of free throws till miss

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**Binomial formulas for mean and standard deviation**

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**In a certain county, 30% of the voters are Republicans**

In a certain county, 30% of the voters are Republicans. How many Republicans would you expect in ten randomly selected voters? What is the standard deviation for this distribution?

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**Geometric formulas for mean and standard deviation**

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**A real estate agent shows a house to prospective buyers**

A real estate agent shows a house to prospective buyers. The probability that the house will be sold to the person is 35%. What is the probability that the agent will sell the house to the third person she shows it to? How many prospective buyers does she expect to show the house to before someone buys the house?

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Binomial Distributions

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