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Binomial Experiments

Binomial Experiments. 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.

caleb-mcfadden
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Binomial Experiments

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  1. Binomial Experiments 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) =qwherep + q = 1 • The central problem is to find the probability of xsuccesses out of n trials. Where x = 0 or 1 or 2 … n. The random variable x is a count of the number of successes in n trials.

  2. 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 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

  3. 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?

  4. Binomial Experiments 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

  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 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

  6. Combination of n values, choosing x There are ways. Find the probability of getting exactly 3 questions correct on the quiz. 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

  7. 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. P(3) = 0.088 P(4) = 0.015 P(5) = 0.001

  8. Binomial Distribution . 3 9 6 . 2 9 4 . 2 3 7 . 0 8 8 . 0 1 5 0 0 1 x P(x) 0 0.237 1 0.396 2 0.264 3 0.088 4 0.015 5 0.001 Binomial Histogram . 4 0 . 3 0 . 2 0 . 1 0 . 0 0 1 2 3 4 5 x

  9. 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 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

  10. 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.

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