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HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Chapter 7 Probability Distributions: Information about the Future

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HAWKES LEARNING SYSTEMS math courseware specialists Probability Distributions: Information about the Future Section 7.1 Types of Random Variables Objectives: To define discrete random variables. To define continuous random variables. To describe probability notation.

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HAWKES LEARNING SYSTEMS math courseware specialists Random variable – a numerical outcome of a random process. Probability distribution – a model which describes a specific kind of random process. Discrete random variable – a random variable which has a countable number of possible outcomes. Continuous random variable – a random variable that can assume any value on a continuous segment(s) of the real number line. Definitions: Probability Distributions: Information about the Future Section 7.1 Types of Random Variables

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HAWKES LEARNING SYSTEMS math courseware specialists Discrete Random Variables: To describe a discrete random variable: State the variable. List all the possible values of the variable. Determine the probabilities of these values. Notation for Random Variables: Capital letters, such as X, will be used to refer to the random variable, while small letters, such as x, will refer to specific values of the random variable. Often the specific values will be subscripted, Probability Distributions: Information about the Future Section 7.1 Types of Random Variables

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HAWKES LEARNING SYSTEMS math courseware specialists Example: Toss a die and observe the outcome of the toss. First list the three steps: State the variable: X = the outcome of the toss of the die. List the possible values: 1, 2, 3, 4, 5, 6. In this case Determine the probability of each value. Probability Distributions: Information about the Future Section 7.1 Types of Random Variables

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HAWKES LEARNING SYSTEMS math courseware specialists Describing a Continuous Random Variable: Time between failure. Calculate the time between installing a brake light in your car and the time the light ceases to work. Defining a continuous random variable is very similar to defining a discrete random variable. Indentify the random variable: X = Time between installation and failure. Indentify the range of values: Between zero and infinity, note X is measured on a continuous scale. Define the probability density: Unknown, but probably would be modeled on historical data and is most likely exponentially distributed. Note: for continuous random variables, we specify probabilities with probability density functions. Probability Distributions: Information about the Future Section 7.1 Types of Random Variables

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HAWKES LEARNING SYSTEMS math courseware specialists Chapter Name Section ## Section Name Probability Distributions: Information about the Future Section 7.2 Discrete Probability Distributions Objectives: To describe the characteristics of a discrete random variable.

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HAWKES LEARNING SYSTEMS math courseware specialists Discrete Probability Distribution – all possible values of a random variable with their associated probabilities. Definition: Characteristics of Discrete Probability Distributions: The sum of all probabilities must equal 1. The probability of any value must be between 0 and 1, inclusively. Probability Distributions: Information about the Future Section 7.2 Discrete Probability Distributions

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HAWKES LEARNING SYSTEMS math courseware specialists Create a probability distribution for X, the number of heads in four tosses of a coin. Solution: To begin, list all possible values of X. Then, to find the probability distribution, we need to calculate the probability of each outcome. Tossing a Coin xP(X=x)Simple Events Probability Distributions: Information about the Future Section 7.2 Discrete Probability Distributions Example:

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HAWKES LEARNING SYSTEMS math courseware specialists The probability distribution for the price of a stock thirty days from now is given below. Find the probability the price of the stock with be greater than $56. Stock Prices xP(X=x) Based on the probability distribution, the probability that the stock price will be more than $56 in thirty days is calculated as follows: Probability Distributions: Information about the Future Section 7.2 Discrete Probability Distributions Example: Solution:

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HAWKES LEARNING SYSTEMS math courseware specialists Probability Distributions: Information about the Future Section 7.3 Expected Value Objectives: To define and describe the expected value of a discrete random variable.

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HAWKES LEARNING SYSTEMS math courseware specialists Expected Value: The expected value of the random variable X is the mean of the random variable X. It is denoted by E(X). Probability Distributions: Information about the Future Section 7.3 Expected Value

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HAWKES LEARNING SYSTEMS math courseware specialists John sells cars. Calculate the expected value of the number of cars John sells per day. Car Sales xP(X=x) Solution: Probability Distributions: Information about the Future Section 7.3 Expected Value Example:

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HAWKES LEARNING SYSTEMS math courseware specialists You are trying to decide between two different investment options. The two plans are summarized in the table below. The left-hand column for each plan gives the potential profits, and the right-hand columns give their respective probabilities. Which plan should you choose? Example : Expected Value Investment AInvestment B ProfitProbabilityProfitProbability $1200.1$ $950.2$800.1 $130.4–$100.2 –$575.1–$250.2 –$1400.2–$690.2 Probability Distributions: Information about the Future Section 7.3 Expected Value

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HAWKES LEARNING SYSTEMS math courseware specialists Solution: It is difficult to determine which plan is better by simply looking at the table. Let’s use the expected value to compare the plans. For Investment A: For Investment B: E(X) = (1200)(.1) + (950)(.2) + (130)(.4) + (–575)(.1) + (–1400)(.2) = – – 280 = $24.50 E(X) = (1500)(.3) + (800)(.1) + (–100)(.2) + (–250)(.2) + (–690)(.2) = – 20 – 50 – 138 = $ Best option Probability Distributions: Information about the Future Section 7.3 Expected Value

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HAWKES LEARNING SYSTEMS math courseware specialists Probability Distributions: Information about the Future Section 7.4 Variance of a Discrete Random Variable Objectives: To define and describe the variance of a discrete random variable.

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HAWKES LEARNING SYSTEMS math courseware specialists Variance of a Discrete Random Variable: The standard deviation is computed by taking the square root of the variance: Variance in investments reflects greater risks. Probability Distributions: Information about the Future Section 7.4 Variance of a Discrete Random Variable

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HAWKES LEARNING SYSTEMS math courseware specialists Variance of a Random Variable Investment AInvestment B ProfitProbabilityProfitProbability $1200.1$ $950.2$800.1 $130.4–$100.2 –$575.1–$250.2 –$1400.2–$690.2 To determine the risk, we need to calculate the variance of each investment. Determine the Risk: Probability Distributions: Information about the Future Section 7.4 Variance of a Discrete Random Variable

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HAWKES LEARNING SYSTEMS math courseware specialists Solution: For Investment A: Variance of a Random Variable Investment A ProfitProbability $ $950.2 $130.4 –$575.1 –$ Probability Distributions: Information about the Future Section 7.4 Variance of a Discrete Random Variable

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HAWKES LEARNING SYSTEMS math courseware specialists Solution: For Investment B: Variance of a Random Variable Investment B ProfitProbability $ $800.1 –$100.2 –$250.2 –$690.2 Probability Distributions: Information about the Future Section 7.4 Variance of a Discrete Random Variable

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HAWKES LEARNING SYSTEMS math courseware specialists Solution: Since in terms of risk Investment B is considered the better option because it carries slightly less risk. Probability Distributions: Information about the Future Section 7.4 Variance of a Discrete Random Variable

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HAWKES LEARNING SYSTEMS math courseware specialists Probability Distributions: Information about the Future Section 7.7 The Binomial Distribution Objectives: To define a Binomial random variable. To calculate probabilities using the Binomial distribution. To calculate the expected value of a Binomial random variable. To calculate the variance of a Binomial random variable.

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HAWKES LEARNING SYSTEMS math courseware specialists Definition: Binomial experiment – a random experiment which satisfies all of the following conditions. i)There are only two outcomes on each trial of the experiment. (One of the outcomes is usually referred to as a success, and the other as a failure.) ii) The experiment consists of n identical trials as described in Condition 1. iii)The probability of success on any one trial is denoted by p and does not change from trial to trial. (Note that the probability of a failure is 1−p and also does not change from trial to trial.) iv)The trials are independent. v)The binomial random variable X is the count of the number of successes in n trials. Probability Distributions: Information about the Future Section 7.7 The Binomial Distribution

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HAWKES LEARNING SYSTEMS math courseware specialists Toss a coin 5 times and observe the number of heads. Define the experiment in terms of our definition of a binomial experiment. i.There are only two outcomes, heads or tails. ii.The experiment will consist of five tosses of a coin. (Hence: n = 5.) iii.The probability of getting a head (success) is and does not change from trial to trial. (Hence: p =.) iv.The outcome of one toss will not affect other tosses. v.The variable of interest is the count of the number of heads in 5 tosses. Probability Distributions: Information about the Future Section 7.7 The Binomial Distribution Example: Solution:

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HAWKES LEARNING SYSTEMS math courseware specialists Toss a coin 4 times and observe the number of heads. Create the probability distribution for the number of heads. Tossing a Coin Events Number of Heads Probability Probability Distributions: Information about the Future Section 7.7 The Binomial Distribution Example: Solution:

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HAWKES LEARNING SYSTEMS math courseware specialists We will define the binomial probability distribution function as follows: represents the number of combinations of n objects taken x at a time (without replacement) and is given by Probability Distributions: Information about the Future Section 7.7 The Binomial Distribution Binomial Probability Distribution Function:

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What is the probability of getting exactly 7 tails in 18 coin tosses? Example: HAWKES LEARNING SYSTEMS math courseware specialists Solution: n = 18, p =.5, x = 7 Probability Distributions: Information about the Future Section 7.7 The Binomial Distribution

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A quality control expert at a large factory estimates that 10% of all batteries produced are defective. If a sample of 20 batteries are taken, what is the probability that no more than 3 are defective? Example: HAWKES LEARNING SYSTEMS math courseware specialists Solution: n = 20, p =.1, x = 3, but this time we need to look at the probability that no more than three are defective, which is P(X ≤ 3). Probability Distributions: Information about the Future Section 7.7 The Binomial Distribution

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HAWKES LEARNING SYSTEMS math courseware specialists Formulas: Binomial expected value and variance can be defined with the following formulas. Example: A quality control expert at a large factory estimates that 10% of all batteries produced are defective. If a sample of 20 batteries is taken, what is the expected value, variance, and standard deviation of the number of defective batteries? Solution: Probability Distributions: Information about the Future Section 7.7 The Binomial Distribution

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HAWKES LEARNING SYSTEMS math courseware specialists Probability Distributions: Information about the Future Section 7.8 The Poisson Distribution Objectives: To define a Poisson random variable. To calculate probabilities using the Poisson distribution.

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HAWKES LEARNING SYSTEMS math courseware specialists Definitions: Poisson distribution – a discrete probability distribution that uses a fixed interval of time or space in which the number of successes are recorded. where In the Poisson distribution Probability Distributions: Information about the Future Section 7.8 The Poisson Distribution

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1.The successes must occur one at a time. 2.Each success must be independent of any other successes. HAWKES LEARNING SYSTEMS math courseware specialists Poisson Distribution Guidelines: When calculating the Poisson distribution, round your answers to four decimal places. Probability Distributions: Information about the Future Section 7.8 The Poisson Distribution

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Suppose that the dial-up Internet connection at your home goes out an average of 0.75 times every hour. If you plan to be connected to the internet for 3 hours one afternoon, what is the probability that you will stay connected the entire time? Assume that the dial-up disconnections follow a Poisson distribution. Example: HAWKES LEARNING SYSTEMS math courseware specialists Solution: x = 0, (0.75)(3) = 2.25 Probability Distributions: Information about the Future Section 7.8 The Poisson Distribution

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A typist averages 1 typographical error per paragraph. If the document has 4 paragraphs, what is the probability that there will be less than 5 mistakes? Example: HAWKES LEARNING SYSTEMS math courseware specialists Solution: x < 5, = This time we need to look at the probability that less than five mistakes will occur, which is P(X < 5). (4)(1) = 4 P(X < 5) = P(X ≤ 4) Probability Distributions: Information about the Future Section 7.8 The Poisson Distribution

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A fast food restaurant averages 2 incorrect orders every 4 hours. What is the probability that they will get at least 3 orders wrong in any given day between 11 AM and 11PM? Assume that fast food errors follow a Poisson distribution. Example: HAWKES LEARNING SYSTEMS math courseware specialists Solution: x ≥ 3, = This time we need to look at the probability that at least three wrong orders will occur, which is P(X ≥ 3). (3)(2) = 6 P(X ≥ 3) = 1 – P(X < 3) = 1 – P(X ≤ 2) Probability Distributions: Information about the Future Section 7.8 The Poisson Distribution

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HAWKES LEARNING SYSTEMS math courseware specialists Objectives: To define a Hypergeometric random variable. To calculate probabilities using the Hypergeometric distribution. To calculate the expected value of a Hypergeometric random variable. To calculate the variance of a Hypergeometric random variable. Probability Distributions: Information about the Future Section 7.9 The Hypergeometric Distribution

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Hypergeometric distribution – a special discrete probability function for problems with a fixed number of dependent trials and a specified number of countable successes. HAWKES LEARNING SYSTEMS math courseware specialists Definitions: When calculating the hypergeometric distribution, round your answers to four decimal places. Probability Distributions: Information about the Future Section 7.9 The Hypergeometric Distribution

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1.Each trial consists of selecting one of the N items in the population and results in either a success or a failure. 2.The experiment consists of n trials. 3.The total number of possible successes in the entire population is A. 4.The trials are dependent. (i.e., selections are made without replacement.) HAWKES LEARNING SYSTEMS math courseware specialists Hypergeometric Distribution Guidelines: Probability Distributions: Information about the Future Section 7.9 The Hypergeometric Distribution

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At the local grocery store there are 50 boxes of cereal on the shelf, half of which contain a prize. Suppose you buy 4 boxes of cereal. What is the probability that 3 boxes contain a prize? Example: HAWKES LEARNING SYSTEMS math courseware specialists A = 25, x = 3, N = 50, n = 4 Probability Distributions: Information about the Future Section 7.9 The Hypergeometric Distribution Solution:

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HAWKES LEARNING SYSTEMS math courseware specialists A produce distributor is carrying 10 boxes of Granny Smith apples and 8 boxes of Golden Delicious apples. If 6 boxes are randomly delivered to one local market, what is the probability that at least 4 of the boxes delivered contain Golden Delicious apples? Example: Solution: A = 8, x ≥ 4, N = 18, n = 6 Probability Distributions: Information about the Future Section 7.9 The Hypergeometric Distribution

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HAWKES LEARNING SYSTEMS math courseware specialists Formulas: Hypergeometric expected value and variance can be defined with the following formulas. Probability Distributions: Information about the Future Section 7.9 The Hypergeometric Distribution

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HAWKES LEARNING SYSTEMS math courseware specialists Example: A produce distributor is carrying 10 boxes of Granny Smith apples and 8 boxes of Golden Delicious apples. If 6 boxes are randomly delivered to one local market, what is the expected value and variance of the distribution? Solution: A = 8, N = 18, n = 6 Probability Distributions: Information about the Future Section 7.9 The Hypergeometric Distribution

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