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**Binomial Probability Distributions**

Created by Tom Wegleitner, Centreville, Virginia Edited by Olga Pilipets, San Diego, California

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Key Concept This section presents a basic definition of a binomial distribution along with notation, and it presents methods for finding probability values. Binomial probability distributions allow us to deal with circumstances in which the outcomes belong to two relevant categories such as acceptable/defective or survived/died. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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Definitions A binomial probability distribution results from a procedure that meets all the following requirements: 1. The procedure has a fixed number of trials. 2. The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.) 3. Each trial must have all outcomes classified into two categories (commonly referred to as success and failure). Binomial probability distributions are important because they allow us to deal with circumstances in which the outcomes belong to TWO categories, such as pass/fall, acceptable/defective, etc. page 208 of Essentials of Statistics, 3rd Edition 4. The probability of a success remains the same in all trials. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Example: # 5-6 p. 215 Identifying Binomial Distributions.**

Determine whether the given procedure results in a binomial distribution. 5. Randomly selecting 12 jurors and recording their nationalities. 6. Randomly selecting 12 jurors and recording whether there is a “no” response when they are asked if they have ever been convicted of felony. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Example: # 7-9 p. 215 Identifying Binomial Distributions.**

7. Treating 50 smokers with Nicorette and asking them how their mouth and throat feel. 8. Treating 50 smokers with Nicorette and recording whether there is a “yes” response when they are asked if they experience any mouth or throat soreness 9. Recording the number of children in 250 families. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Notation for Binomial Probability Distributions**

S and F (success and failure) denote two possible categories of all outcomes; p and q will denote the probabilities of S and F, respectively, so P(S) = p (p = probability of success) The word success is arbitrary and does not necessarily represent something GOOD. If you are trying to find the probability of deaths from hang-gliding, the ‘success’ probability is that of dying from hang-gliding. Remind students to carefully look at the probability (percentage or rate) provided and whether it matches the probability desired in the question. It might be necessary to determine the complement of the given probability to establish the ‘p’ value of the problem. Page 208 of Essentials of Statistics, 3rd Edition. P(F) = 1 – p = q (q = probability of failure) Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Notation (cont) n denotes the number of fixed trials.**

x denotes a specific number of successes in n trials, so x can be any whole number between 0 and n, inclusive. p denotes the probability of success in one of the n trials. q denotes the probability of failure in one of the n trials. Page 208 of Essentials of Statistics, 3rd Edition. P(x) denotes the probability of getting exactly x successes among the n trials. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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Important Hints Be sure that x and p both refer to the same category being called a success. When sampling without replacement, consider events to be independent if n < 0.05N. Page of Essentials of Statistics, 3rd Edition. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Methods for Finding Probabilities**

We will now discuss three methods for finding the probabilities corresponding to the random variable x in a binomial distribution. Page 210 of Essentials of Statistics, 3rd Edition. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Example: #14 p.215 Finding Probabilities When Guessing Answers.**

A test consists of multiple-choice questions, each having four possible answers (a, b, c, d), one of which is correct. Assume that you guess the answers to six such questions. a) Use the multiplication rule to find the probability that the first two guesses are wrong and the last four guesses are correct. That is, find P(WWCCCC), where C denotes a correct answer and W denotes a wrong answer. b) Beginning with WWCCCC, make a complete list of the different possible arrangements of two wrong answers and four correct answers, then find the probability for each entry in the list. c) Based on the preceding results, what is the probability of getting exactly four correct answers when six guesses are made? Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Method 1: Using the Binomial**

Probability Formula P(x) = • px • qn-x (n – x )!x! n ! for x = 0, 1, 2, . . ., n where n = number of trials x = number of successes among n trials p = probability of success in any one trial q = probability of failure in any one trial (q = 1 – p) page 210 of Essentials of Statistics, 3rd Edition Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Example 3: Using the Binomial Probability Formula.**

It can be very difficult to make sales by means of telephone transactions. Suppose that it is known that only 10% of all business calls result in a sale. What is the probability that out of the next 15 such calls, exactly 3 will result in a sale? Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Method 2: Using Table A-1 in Appendix A**

Part of Table A-1 is shown below. With n = 12 and p = 0.80 in the binomial distribution, the probabilities of 4, 5, 6, and 7 successes are 0.001, 0.003, 0.016, and respectively. Page 211 of Essentials of Statistics, 3rd Edition. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Example 4: Using A-1 table.**

There are 12 students in a study group. Forty percent of the students are female. The instructor randomly selects 5 students to attend a lecture on quality control and industrial statistics. a) What is the probability that at most 2 of the 5 students selected will be female? b) What is the probability that more than 3 students selected will be female? Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Method 3: Using Technology**

STATDISK, Minitab, Excel and the TI-83 Plus calculator can all be used to find binomial probabilities. Excel TI-83 Plus calculator Page 212 of Essentials of Statistics, 3rd Edition. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Method 3: Using Technology**

To view a list of probability values for given n and p on TI-83 Plus calculator Press 2-nd VARS to get DISTR Scroll down, select binompdf Complete the entry binompdf(n,p) with the given values for n and p Press ENTER. You can view the list of values for all x =0,1,2,3,…,n Store the values into list 2 by pressing STOL2 Fill in L1 with x values Page 212 of Essentials of Statistics, 3rd Edition. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Method 3: Using Technology**

To view probability value for a specified x with the given n and p on TI-83 Plus calculator Press 2-nd VARS to get DISTR Scroll down, select binompdf Complete the entry binompdf(n,p,x) with the given values for x, n, and p Press ENTER. Page 212 of Essentials of Statistics, 3rd Edition. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Method 3: Using Technology**

To display a cumulative probability for all x’s from x=0 through a specified x with the given n and p on TI-83 Plus calculator Press 2-nd VARS to get DISTR Scroll down, select binomcdf Complete the entry binomcdf(n,p,x) with the given values for x, n, and p Press ENTER. Page 212 of Essentials of Statistics, 3rd Edition. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Example: Acceptance Sampling**

The Medassist Pharmaceutical Company receives large shipments of aspirin tablets and uses this acceptance sampling plan: Randomly select and test 24 tablet, then accept the whole batch if there is only one or none that doesn’t meet the required specifications. If a particular shipment of thousands of aspirin tablets actually has a 4% rate of defects, what is the probability that this whole shipment will be accepted? Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Strategy for Finding Binomial Probabilities**

Use computer software or a TI-83 Plus calculator if available. If neither software nor the TI-83 Plus calculator is available, use Table A-1, if possible. If neither software nor the TI-83 Plus calculator is available and the probabilities can’t be found using Table A-1, use the binomial probability formula. Page 213 of Essentials of Statistics, 3rd Edition Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Rationale for the Binomial Probability Formula**

P(x) = • px • qn-x n ! (n – x )!x! The number of outcomes with exactly x successes among n trials The ‘counting’ factor of the formula counts the number of ways the x successes and (n-x) failures can be arranged - i.e.. the number of arrangements (Review section 4-6, page 176). Discussion is on page 213 of Essentials of Statistics, 3rd Edition. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Binomial Probability Formula**

P(x) = • px • qn-x n ! (n – x )!x! Number of outcomes with exactly x successes among n trials The probability of x successes among n trials for any one particular order The remaining two factors of the formula will compute the probability of any one arrangement of successes and failures. This probability will be the same no matter what the arrangement is. The three factors multiplied together give the correct probability of ‘x’ successes. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Recap In this section we have discussed:**

The definition of the binomial probability distribution. Notation. Important hints. Three computational methods. Rationale for the formula. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Mean, Variance, and Sdandard Deviation for the Binomial Probability Distributions**

Created by Tom Wegleitner, Centreville, Virginia Edited by Olga Pilipets, San Diego, California

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Key Concept In this section we consider important characteristics of a binomial distribution including center, variation and distribution. That is, we will present methods for finding its mean, variance and standard deviation. As before, the objective is not to simply find those values, but to interpret them and understand them. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**For Any Discrete Probability Distribution: Formulas**

Mean µ = [x • P(x)] Variance 2= [ x2 • P(x) ] – µ2 Std. Dev = [ x2 • P(x) ] – µ2 These formulas were introduced in section 5-2, pages These formulas will produce the mean, standard deviation, and variance for any probability distribution. The difficult part of these formulas is that one must have all the P(x) values for the random variables in the distribution. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Binomial Distribution: Formulas**

Std. Dev. = n • p • q Mean µ = n • p Variance 2 = n • p • q Where n = number of fixed trials p = probability of success in one of the n trials q = probability of failure in one of the n trials The obvious advantage to the formulas for the mean, standard deviation, and variance of a binomial distribution is that you do not need the values in the distribution table. You need only the n, p, and q values. A common error students tend to make is to forget to take the square root of (n)(p)(q) to find the standard deviation, especially if they used L1 and L2 lists in a graphical calculator to find the standard deviation with non-binomial distributions. The square root is built into the programming of the calculator and students do not have to remember it. These formulas do require the student to remember to take the square root of the three factors. Page 219 of Essentials of Statistics, 3rd Edition Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Interpretation of Results**

It is especially important to interpret results. The range rule of thumb suggests that values are unusual if they lie outside of these limits: Maximum usual values = µ + 2 Minimum usual values = µ – 2 Page 219 of Essentials of Statistics, 3rd Edition. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Example: #10 p 222. Guessing Answers**

Several economics students are unprepared for a multiple –choice quiz with 25 questions, and all of their answers are guesses. Each question has five possible answers, and only one of them is correct. a) Find the mean and Standard deviation for the number of correct answers for such students. b) Would it be unusual for a student to pass by guessing and getting at least 15 correct answers? Why or why not? Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Recap In this section we have discussed:**

Mean,variance and standard deviation formulas for the any discrete probability distribution. Mean,variance and standard deviation formulas for the binomial probability distribution. Interpreting results. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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