2. Discrete Probability Distributions Elementary Statistics Larson Farber x = number of on time arrivals x = number of points scored in a game x = number of employees reaching quota x = number of correct answers Chapter 4

3. A random variable, x is the numerical outcome of a probability experiment x = The number of people in a car. x = The gallons of gas bought in a week. x = The time it takes to drive from home to school x = The number of trips to school you make per week A random variable is discrete if the number of possible outcomes is finite or countable. Discrete random variables are determined by a count. A random variable is continuous if it can take on any value within an interval. The possible outcomes cannot be listed. Continuous random variables are determined by a measure. Random Variables Two types of random variables

4. Types of Random Variables x = The number of people in a car. x = The gallons of gas bought in a week. x = The time it takes to drive from home to school x = The number of trips to school you make per week Identify each random variable as discrete or continuous. Discrete-you count the number of people in a car 0, 1, 2, 3… Possible values can be listed. Continuous-you measure the gallons of gas. You cannot list the possible values. Continuous-you measure the amount of time. The possible values cannot be listed. Discrete-you count the number of trips you make. The possible numbers can be listed.

5. Discrete Probability Distributions A discrete probability distribution lists each possible value of the random variable, together with its probability. A survey asks a sample of families how many vehicles each owns. number of vehicles Properties of a probability distribution Each probability must be between 0 and 1, inclusive. The sum of all probabilities is 1.

6. Probability Histogram The height of each bar corresponds to the probability of x. When the width of the bar is 1, the area of each bar corresponds to the probability the value of x will occur. 0123

7. Mean, Variance and Standard Deviation The variance of a discrete probability distribution is: The standard deviation of a discrete probability distribution is: The mean of a discrete probability distribution is:

8. Mean (Expected value) Multiply each value times its probability. Add the products The expected value (the mean) is 1.763 vehicles. Calculate the mean

9. Variance and Standard Deviation The standard deviation is 0.775 vehicles. The mean is 1.763 vehicles. Calculate the variance Calculate the standard deviation μμμ 2 variance

10. 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 and the probability of failure is q. P(S) = p P(F) =q 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. Binomial Experiments Characteristics of a Binomial Experiment x is a count of the number of successes in n trials.

11. 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 Quiz

12. Quiz Results 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? The correct answers to the quiz are: 1. d 2. a 3. b 4. c 5. b

13. 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. 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. n = 8p = 1/3q = 2/3x = 5 n = 7p = 0.80 q = 0.20 x = 6 Binomial Experiments

14. Find the probability of getting exactly 3 questions correct on the quiz. Denote 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 SFFSS There areways. 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

15. 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) =0.088P(4) =0.015P(5) =0.001

16. Binomial Distribution xP(x) 00.237 10.396 20.264 30.088 40.015 50.001 Binomial Histogram x

17. Probabilities 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 xP(x) 00.237 10.396 20.264 30.088 40.015 50.001

18. Calculate the mean, variance and standard deviation xP(x)xP(x) 00.2370.000 10.396 20.2640.527 30.0880.264 40.0150.059 50.0010.005 The expected value (mean) is 1.25, the variance is 0.9375, and the standard deviation Is 0.968 -1.25 1.5625 0.371 -0.25 0.0625 0.025 0.75 0.5625 0.148 1.75 3.0625 0.269 2.75 7.5625 0.111 3.7514.0625 0.014

19. Parameters for a Binomial Experiment Use the binomial formulas to find the mean, variance and standard deviation for the distribution of correct answers on the quiz. Mean: Variance : Standard deviation: