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**Chapter 5 Discrete Probability Distributions**

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Chapter 5 Overview Introduction 5-1 Probability Distributions**

5-2 Mean, Variance, Standard Deviation, and Expectation 5-3 The Binomial Distribution Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Chapter 5 Objectives Construct a probability distribution for a random variable. Find the mean, variance, standard deviation, and expected value for a discrete random variable. Find the exact probability for X successes in n trials of a binomial experiment. Find the mean, variance, and standard deviation for the variable of a binomial distribution. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**5.1 Probability Distributions**

A random variable is a variable whose values are determined by chance. The variable includes only whole numbers ( no fractions or irrational numbers). Values of this variable include 0 , 1, 2, 3, … A discrete probability distribution consists of the values a random variable can assume and the corresponding probabilities of the values. The sum of the probabilities of all events in a sample space add up to 1. Each probability is between 0 and 1, inclusively. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Examples of random variables:**

1. the number of speeding tickets issued on a certain stretch of I 95 S. 2. the number of heads which appear when 4 dimes are tossed. 3. The number of passes completed in a game by a quarterback 4. The number of defective parts in a lot of 1000. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Continuous random variables**

Temperature: Range degrees Can assume an infinite number of values Examples: weight, height, temperature, time.

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**Chapter 5 Discrete Probability Distributions**

Section 5-1 Example 5-1 Page #262 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Example 5-1: Rolling a Die**

Construct a probability distribution for rolling a single die. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Chapter 5 Discrete Probability Distributions**

Section 5-1 Example 5-2 Page #262 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Discrete probability distribution**

-consists of a correspondence between the values of the random variable and its associated probability. The correspondence can be shown in a table. Properties:

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**Example 5-2: Tossing Coins**

Represent graphically the probability distribution for the sample space for tossing three coins. . Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Discrete probability distribution**

A bag contains three red checkers and two black checkers. Two checkers are drawn in succession and without replacement from this bag. Let x represent the the number of red checkers drawn in two attempts. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Number of red checkers A) What are the values that x can take on?**

Answer: x = 0, 1, 2 B) Find the probability distribution of x, the number of red checkers. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Table X P(x) 0.1 1 0.6 2 0.3 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Probability Distribution?**

C) Does the distribution satisfy the requirements of a discrete probability distribution? Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Mean of a probability distribution**

Toss three coins and let x represent the number of heads that appear. Recall the sample space: {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} X = 0, 1, 1, 1, 2, 2, 2, 3 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Mean Notice the the outcomes of x = 1 and x = 2 occur three times each, while the outcomes x = 0 and x = 3 occur once each: Notice here we can find the mean by multiplying the values of x by the probability that x occurs. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Mean x p(x) xp(x) 0 1/8 0(1/8)= 0 1 3/8 1(3/8)=3/8 2 3/8 2(3/8)=6/8**

0 1/8 0(1/8)= 0 1 3/8 1(3/8)=3/8 2 3/8 2(3/8)=6/8 3 1/8 3(1/8)=3/8 sum = 3/2 = 1.5 Thus, mean number of heads = 1.5 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Mean number of red checkers**

Recall the problem of selecting two checkers from a bag containing 2 black and 3 red checkers X P(x) X*P(x) 0.1 1 0.6 2 0.3 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**5-2 Mean, Variance, Standard Deviation, and Expectation**

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Mean, Variance, Standard Deviation, and Expectation**

Rounding Rule The mean, variance, and standard deviation should be rounded to one more decimal place than the outcome X. When fractions are used, they should be reduced to lowest terms. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Chapter 5 Discrete Probability Distributions**

Section 5-2 Example 5-5 Page #268 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Example 5-5: Rolling a Die**

Find the mean of the number of spots that appear when a die is tossed. . Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Chapter 5 Discrete Probability Distributions**

Section 5-2 Example 5-8 Page #269 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Example 5-8: Trips of 5 Nights or More**

The probability distribution shown represents the number of trips of five nights or more that American adults take per year. (That is, 6% do not take any trips lasting five nights or more, 70% take one trip lasting five nights or more per year, etc.) Find the mean. . Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Example 5-8: Trips of 5 Nights or More**

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Chapter 5 Discrete Probability Distributions**

Section 5-2 Example 5-9 Page #270 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Example 5-9: Rolling a Die**

. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Chapter 5 Discrete Probability Distributions**

Section 5-2 Example 5-11 Page #271 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Example 5-11: On Hold for Talk Radio**

A talk radio station has four telephone lines. If the host is unable to talk (i.e., during a commercial) or is talking to a person, the other callers are placed on hold. When all lines are in use, others who are trying to call in get a busy signal. The probability that 0, 1, 2, 3, or 4 people will get through is shown in the distribution. Find the variance and standard deviation for the distribution. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Example 5-11: On Hold for Talk Radio**

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Example 5-11: On Hold for Talk Radio**

A talk radio station has four telephone lines. If the host is unable to talk (i.e., during a commercial) or is talking to a person, the other callers are placed on hold. When all lines are in use, others who are trying to call in get a busy signal. Should the station have considered getting more phone lines installed? Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Example 5-11: On Hold for Talk Radio**

No, the four phone lines should be sufficient. The mean number of people calling at any one time is 1.6. Since the standard deviation is 1.1, most callers would be accommodated by having four phone lines because µ + 2σ would be (1.1) = = 3.8. Very few callers would get a busy signal since at least 75% of the callers would either get through or be put on hold. (See Chebyshev’s theorem in Section 3–2.) Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Expectation The expected value, or expectation, of a discrete random variable of a probability distribution is the theoretical average of the variable. The expected value is, by definition, the mean of the probability distribution. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Chapter 5 Discrete Probability Distributions**

Section 5-2 Example 5-12 Page #272 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Example 5-12: Winning Tickets**

One thousand tickets are sold at $1 each for a color television valued at $350. What is the expected value of the gain if you purchase one ticket? Win Lose Gain X $349 -$1 Probability P(X) Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Example 5-13: Winning Tickets**

One thousand tickets are sold at $1 each for four prizes of $100, $50, $25, and $10. After each prize drawing, the winning ticket is then returned to the pool of tickets. What is the expected value if you purchase two tickets? Alternate Approach Gain X Probability P(X) $100 $50 $25 $10 $0 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Expected value Life insurance. A person pays $100 for a one year term life insurance policy. The probability that this person will live one more year is The policy pays his beneficiaries $3000 upon the death of the policy holder. What is the expected value of x, the amount of money the insurance company expects to make? Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Expected Value X P(x) X*P(x) 100 0.97 97 -2900 0.03 -87**

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Expected Value There are two outcomes: The insured lives for one more year or dies within one year. If the person lives for one more year, the insurance company gains $100. If the person dies within a year, the beneficiaries receive $2900. (The premium of $100 is subtracted from the benefit. X P(x) X*P(x) 100 0.97 97 -2900 0.03 -87

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**5-3 The Binomial Distribution**

Many types of probability problems have only two possible outcomes or they can be reduced to two outcomes. Examples include: when a coin is tossed it can land on heads or tails, when a baby is born it is either a boy or girl, etc. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**The Binomial Distribution**

The binomial experiment is a probability experiment that satisfies these requirements: Each trial can have only two possible outcomes— success or failure. There must be a fixed number of trials. The outcomes of each trial must be independent of each other. The probability of success must remain the same for each trial. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Notation for the Binomial Distribution**

P(S) P(F) p q P(S) = p n X Note that X = 0, 1, 2, 3,...,n The symbol for the probability of success The symbol for the probability of failure The numerical probability of success The numerical probability of failure and P(F) = 1 – p = q The number of trials The number of successes Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Introduction to Binomial Probability**

Suppose there is a 0.30 probability that a customer walks in a store and buys at least one product. Find the probability that two out of three customers will buy at least one product. This is a binomial experiment. Why? 1) Fixed number of trials (3) 2) Two outcomes 3) Trials independent Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Binomial experiment There are C(3,2) that any two of the three customers will purchase at least one product. (Check this by using a tree diagram). C(3,2) = For each of the three outcomes, the probability is 0.3(0.3)(0.7) = Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**This result is generalized on the next slide.**

Binomial probability Thus P(x = 2) = This result is generalized on the next slide. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**The Binomial Distribution**

In a binomial experiment, the probability of exactly X successes in n trials is Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Chapter 5 Discrete Probability Distributions**

Section 5-3 Example 5-16 Page #280 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Example 5-16: Survey on Doctor Visits**

A survey found that one out of five Americans say he or she has visited a doctor in any given month. If 10 people are selected at random, find the probability that exactly 3 will have visited a doctor last month. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Chapter 5 Discrete Probability Distributions**

Section 5-3 Example 5-17 Page #281 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Example 5-17: Survey on Employment**

A survey from Teenage Research Unlimited (Northbrook, Illinois) found that 30% of teenage consumers receive their spending money from part-time jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them will have part-time jobs. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Chapter 5 Discrete Probability Distributions**

Section 5-3 Example 5-18 Page #281 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Example 5-18: Tossing Coins**

A coin is tossed 3 times. Find the probability of getting exactly two heads, using Table B. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**The Binomial Distribution**

The mean, variance, and standard deviation of a variable that has the binomial distribution can be found by using the following formulas. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Chapter 5 Discrete Probability Distributions**

Section 5-3 Example 5-23 Page #284 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Example 5-23: Likelihood of Twins**

The Statistical Bulletin published by Metropolitan Life Insurance Co. reported that 2% of all American births result in twins. If a random sample of 8000 births is taken, find the mean, variance, and standard deviation of the number of births that would result in twins. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Using the TI83+ Suppose n = 10 , p = 0.3 and x = 3.**

We can calculate the binomial probability using the TI 83+ calculator by following the steps below: 1) Enter 2nd 2) DISTR 3) Option 10 (binompdf( 4. enter binompdf(10, 0.3, 3) =0.2668 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**Using the TI 83+ to calculate cumulative binomial probability**

Let n = 10, p = 0.3 and x is less than or equal to 3. So what is calculated is P(x =0) + P(x = 1) + P(x = 2) + p(x = 3). 1. 2nd 2. DISTR 3) Option A (binomcdf(10, 0.3, 3)) Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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