Binomial Distributions Section 4.2 Binomial Distributions

Binomial Experiments The random variable x is a count of the Characteristics of a Binomial Experiment There are a fixed number of trials. (n) The n trials are independent and repeated under identical conditions. Each trial has 2 outcomes, S = Success or F = Failure. The probability of success on a single trial is p. P(S) = p The probability of failure is q. P(F) =q where p + q = 1 The central problem is to find the probability of x successes out of n trials. Where x = 0 or 1 or 2 … n. Examples: roll a die eight times and count the number of times a 4 appeared. Throw 12 basketball free shots and count the number of times the ball went through the basket. Make 15 sales calls and count the number of sales made. The random variable x is a count of the number of successes in n trials.

Guess the Answers 1. What is the 11th digit after the decimal point for the irrational number e? (a) 2 (b) 7 (c) 4 (d) 5 2. What was the Dow Jones Average on February 27, 1993? (a) 3265 (b) 3174 (c) 3285 (d) 3327 3. How many students from Sri Lanka studied at U.S. universities from 1990-91? (a) 2320 (b) 2350 (c) 2360 (d) 2240 Have students take this quiz explaining that they should simply guess each answer. 4. How many kidney transplants were performed in 1991? (a) 2946 (b) 8972 (c) 9943 (d) 7341 5. How many words are in the American Heritage Dictionary? (a) 60,000 (b) 80,000 (c) 75,000 (d) 83,000

Quiz Results The correct answers to the quiz are: 1. d 2. a 3. b 4. c 5. b Count the number of correct answers. Let the number of correct answers = x. Why is this a binomial experiment? What are the values of n, p and q? What are the possible values for x? This example helps students appreciate the concept of expected value. This is a binomial experiment with a trial being answering a question. There are 5 questions, so n =5. Each question has 4 choices, so p = ¼ or .25. The value of q = ¾ or .75. Possible values of x are {0, 1, 2, 3, 4, 5}.

Binomial Experiments n = 8 p = 1/3 q = 2/3 x = 5 q = 0.20 n = 7 A multiple choice test has 8 questions each of which has 3 choices, one of which is correct. You want to know the probability that you guess exactly 5 questions correctly. Find n, p, q, and x. n = 8 p = 1/3 q = 2/3 x = 5 A doctor tells you that 80% of the time a certain type of surgery is successful. If this surgery is performed 7 times, find the probability exactly 6 surgeries will be successful. Find n, p, q, and x. q = 0.20 n = 7 p = 0.80 x = 6

Try It Yourself Decide whether the following is a Binomial experiment. If it is specify the values of n, p, and q And list the possible values of the random variable x. If it is not explain why. You take a multiple choice quiz that consists of 10 questions. Each question has 4 possible answers only one of which is correct. To complete the quiz, you randomly guess the answer to each question. The random variable represents the number of correct answers. Identify a trial of the experiment and what is a success. b. Decide if the experiment satisfies the four conditions of a binomial experiment. c. Make a conclusion and identify n,p, q and the possible values of x. 1. A Trial: answering a question. Success answering the question correctly. b. Yes c. n=10, p=1/4 = 0.25, q = ¾ = 0.75, x = 0,1,2,3,…10

Is this a binomial experiment? 5. Cyanosis is the condition of having blush skin due to insufficient oxygen to the blood. A bout 80% of the babies born with Cyanosis recover fully. A hospital is caring for 5 babies born with Cyanosis. The random variable represents the number of babies born that recover fully. Is this a binomial experiment? If so what is n, p, q and what are the random values of x? a.) Yes n =5, p = 4/5 = 80%, q = 1/5 = 20% , x = 0,1,2…5

Binomial Probabilities Find the probability of getting exactly 3 questions correct on the quiz. Write the first 3 correct and the last 2 wrong as SSSFF P(SSSFF) = (.25)(.25)(.25)(.75)(.75) = (.25)3(.75)2 = 0.00879 Since order does not matter, you could get any combination of three correct out of five questions. List these combinations. SSSFF SSFSF SSFFS SFFSS SFSFS FFSSS FSFSS FSSFS SFSSF FFSSF Have students list as many of the combinations of 3 successes and 2 failures that they can. If you did not cover combinations in the last chapter, discuss how to calculate them here. Each of these 10 ways has a probability of 0.00879. P(x = 3) = 10(0.25)3(0.75)2 = 10(0.00879) = 0.0879

Binomial Probability Formula In a Binomial experiment the probability of exactly x successes in n trials is P(X) = n C x px qn-x = n! = px qn-x (n-x)!x!

Combination of n values, choosing x There are ways. Find the probability of getting exactly 3 questions correct on the quiz. Have students list as many of the combinations of 3 successes and 2 failures that they can. If you did not cover combinations in the last chapter, discuss how to calculate them here. Each of these 10 ways has a probability of 0.00879. P(x = 3) = 10(0.25)3(0.75)2= 10(0.00879)= 0.0879

Binomial Probabilities In a binomial experiment, the probability of exactly x successes in n trials is Use the formula to calculate the probability of getting none correct, exactly one, two, three, four correct or all 5 correct on the quiz. Show students the factorial key on the calculator. P(3) = 0.088 P(4) = 0.015 P(5) = 0.001

Binomial Distribution x P(x) 0 0.237 1 0.396 2 0.264 3 0.088 4 0.015 5 0.001 Binomial Histogram . 3 9 6 . 4 . 3 . 2 9 4 . 2 3 7 . 2 Discuss the areas of each rectangle and the connection to the probability the particular random variable occurs. . 1 . 8 . 1 5 . 1 1 2 3 4 5 x

Probabilities x P(x) 0 0.237 1 0.396 2 0.264 3 0.088 4 0.015 5 0.001 1. What is the probability of answering either 2 or 4 questions correctly? 2. What is the probability of answering at least 3 questions correctly? 3. What is the probability of answering at least one question correctly? P( x = 2 or x = 4) = 0.264 + 0.015 = 0. 279 Explain that the table in the back of the text gives selected values for binomial probability distributions. P(x  3) = P( x = 3 or x = 4 or x = 5) = 0.088 + 0.015 + 0.001 = 0.104 P(x  1) = 1 - P(x = 0) = 1 - 0.237 = 0.763

Parameters for a Binomial Experiment Mean: Variance: Standard deviation: Use the binomial formulas to find the mean, variance and standard deviation for the distribution of correct answers on the quiz. Have students compare these answers to the ones obtained from the general formula.

9.) Find the mean, Variance, and Standard deviation of the Binomial distribution when n =100, p = 0.4 Mean = = 100 (0.4) = 40 Variance = q= 1- p = 1- 0.4 = .6 100(.4)(.6) = 24 Stand. Deviation = = =4.89

Home wk: 1-12 all Pg. 193-194 Day 2 Home wk: 13-26 all Pg. 194-196

As n increases the distribution becomes more symmetric. Answers to 1.) (a) p= 0.50 (b) p = 0.20, (c ) p=0.80 2.) (a) p=0.75, (b) p = 0.50, ( c ) p= 0.25 3.) (a) n= 12 , (b) n = 4, ( c ) n= 8 As n increases the distribution becomes more symmetric. 4.) (a) n= 10 , (b) n = 15, ( c ) n= 5 As n increases the distribution becomes more symmetric See #5 Above 6.) a.) Yes Binomial Experiment n =18, p = 74%, q = 26% , x = 0,1,2…18 7.) No this is not a binomial Experiment there are more than 2 possible outcomes for each trial. 8.) No this is not a binomial Experiment because the probability for success is not the same for each trial. 9.) 40;24;4.9 (10) 60.8; 12.16; 3.487 11.) 22.08; 18.547; 4.307 (12.) 215.46; 79.720; 8.929