Section 5-3 Binomial Probability Distributions

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1 Section 5-3 Binomial Probability Distributions

2 Key Concept This section presents a basic definition of a binomial distribution along with notation, and 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.

3 Simplify (a + b) 2 a 2 + 2ab + b 2 Simplify (a + b) 3 a 3 + 3a 2 b + 3ab 2 + b 3 Simplify (a + b) 4 a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4

4 An experiment is called a Binomial Experiment (Bernoulli experiment) with n trials if all of the following properties are true: 1. The experiment is repeated n times. 2. Each trial has two outcomes: success or failure. 3. The trials are independent. 4. The probability of a success remains the same in all trials.

5 Ex. 1 Determine whether or not the following are binomial experiments. (c) You roll a die 10 times. What is the probability that you get a six as many times as you get a four? 1. n = 10 (the number of trials) 2. A success is.?????????? 3. p = (the probability of a success) 4. x = (the number of successes) We cannot classify outcomes on the trials as successes and failures. On a single roll of the die, would you consider a six a success, a four a success,....?

6 Determine whether or not the following are binomial experiments. You roll a die 10 times and want to know the probability that a five will come up seven times. 1. n = 10 (the number of trials) 2. A success is. getting a five on a roll of the die 1 / 6 3. p = (the probability of a success) 4. x = (the number of successes) 7

7 Determine whether or not the following are binomial experiments. You draw eight cards from a deck without replacement and want the probability that you draw four spades. 1. n = 8 (the number of trials) 2. A success is. drawing a spade 3. p =??? (the probability of a success) changes on each draw, not the same. 4. x = (the number of successes)

8 Notation for Binomial Probability Distributions S and F (success and failure) denote the 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) P( F) 1 p q (q = probability of failure)

9 To check if these properties are true on a particular problem, you should be able to fill in the blanks in the following checklist: 1. n = (the number of trials in the experiment) 2. x = of successful outcomes. 3. p = (the probability of a success on any one of the trials) 4. q = (the probability of a failure on any one trial.)

10 There are ten multiple choice questions on a test. Each question has four answers. If you randomly guess at the answers, what is the probability of guessing seven correctly (thus getting a C on the test)? 1. n = 10 (the number of trials in the experiment) 2. x = 7 of successful outcomes. 1 / 4 3. p = (the probability of a success on any one of the trials) 3 / 4 4. q = (the probability of a failure on any one trial.)

11 You roll a die 10 times and want to know the probability that a five will come up seven times. 1. n = 10 (the number of trials in the experiment) 2. x = 7 of successful outcomes. 1/6 3. p = (the probability of a success on any one of the trials) 5/6 4. q = (the probability of a failure on any one trial.)

12 You draw eight cards from a deck with replacement and want the probability that you draw four spades. 1. n = 8 (the number of trials in the experiment) 2. x = 4 of successful outcomes. 3. p = 13/52 (the probability of a success on any one of the trials) 4. q = (the 39/52 probability of a failure on any one trial.)

13 Homework Pg odd

14 Section 5-3 II Binomial Probability Distributions

15 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.

16 Methods for Finding Probabilities We will now discuss three methods for finding the probabilities corresponding to the random variable x in a binomial distribution.

17 Method 1: Using the Binomial Probability Formula n! x P( x) p q ( n x)! x! n x n C x p x q n x 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)

18 Rationale for the Binomial Probability Formula n! P( x) p x q ( n x)! x! n x The number of outcomes with exactly x successes among n trials

19 Binomial Probability Formula n! P( x) p x q ( n x)! x! n x Number of outcomes with exactly x successes among n trials The probability of x successes among n trials for any one particular order

20 You draw eight cards from a deck with replacement and want the probability that you draw four spades. 8 C 4 (13/52) 4 (39/52) 4 = n = 8 (the number of trials in the experiment) 2. x = 4 of successful outcomes. 3. p = 13/52 (the probability of a success on any one of the trials) 4. q = (the 39/52 probability of a failure on any one trial.)

21 There are ten multiple choice questions on a test. Each question has four answers. If you randomly guess at the answers, what is the probability of guessing seven correctly? P(getting 7 correct answers out of 10) 10 C 7 (0.25) 7 (0.75) 3 binompdf(10,.25,7)

22 A coin is weighted so that it lands on heads 70% of the time. This coin is flipped eight times. Find the probability of getting six heads on the eight tosses. P(getting 6 heads in 8 tosses) C(8, 6) (0.70) 6 (0.30) 2 binompdf(8,0.70,6)

23 A stats test consists of multiple choice questions, each having 4 possible answers (a, b, c, d), 1 of which is correct. Assume that you guess the answers to 6 questions. a) use the multiplication rule to find the probability that the first 2 guesses are wrong and the last 4 guesses are correct. That is, find P(W, W, C, C, C, C), where C denotes a correct answer and W denotes a wrong answer

24 A stats test consists of multiple choice questions, each having 4 possible answers (a, b, c, d), 1 of which is correct. Assume that you guess the answers to 6 questions. b) Find the probability of getting exactly 4 correct answers when 6 guesses are made. P(4 out of 6 correct answers) C(6, 4) (0.25) 4 (0.74) 2 binompdf(6,0.25,4)

25 In exercises 15-20, you may assume that the problem given yields a binomial distribution with a trial repeated n times. You may either use Table A-1 or the Ti-84 to find the probability of x successes given the probability p of success on a given trial. n = 5, x = 1, p = 0.95 C(5, 1) (0.95) 1 (0.05) 4 binompdf(5,0.95,1) 0+

26 Find the probability that at least 3 of the 5 donors have Group O blood. If at least 3 Group O donors are needed, is it very likely that at least 3 will be obtained? (p = 0.45) C(5, 3) (0.45) 3 (0.55) 2 + C(5, 4) (0.45) 4 (0.55) 1 + C(5, 5) (0.45) 5 (0.55) 0

27 Method 2: Using Technology STATDISK, Minitab, Excel, SPSS, SAS and the TI-83/84 Plus calculator can be used to find binomial probabilities. STATDISK MINITAB

28 Method 2: Using Technology STATDISK, Minitab, Excel and the TI-83 Plus calculator can all be used to find binomial probabilities. EXCEL TI-83 PLUS Calculator

29 Method 3: 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.

30 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.

31 Recap In this section we have discussed: The definition of the binomial probability distribution. Notation. Important hints. Three computational methods. Rationale for the formula.

32

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