2. Larson/Farber Ch. 4 PROBABILITY DISTRIBUTIONS Statistics Chapter 6 For Period 3, Mrs Pullo’s class x = number of on time arrivals x = number of points scored in a game x = number of employees reaching sales quota x = number of correct answers

3. Larson/Farber Ch. 4 2 RANDOM VARIABLES 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

4. Larson/Farber Ch. 4 3 TYPES OF RANDOM VARIABLES 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.

5. Larson/Farber Ch. 4 TYPES OF RANDOM VARIABLES 4 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.

6. Larson/Farber Ch. 4 A probability distribution provides the possible values of the random variable and their corresponding probabilities. A probability distribution can be in the form of a table, graph or mathematical formula.

7. Larson/Farber Ch. 4 6 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.

8. Larson/Farber Ch. 4 EXAMPLE Identifying Probability Distributions Is the following a probability distribution?

9. Larson/Farber Ch. 4 ANSWER: NO 0.16 + 0.18 + 0.22 + 0.10 + 0.3 + 0.01 = 0.97 <1, Not a probability distribution.

10. Larson/Farber Ch. 4 EXAMPLE Identifying Probability Distributions Is the following a probability distribution?

11. Larson/Farber Ch. 4 ANSWER: YES 0.16 + 0.18 + 0.22 + 0.10 + 0.3 + 0.04 = 1 It is a probability distribution

12. Larson/Farber Ch. 4 PROBABILITY HISTOGRAM 11 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

13. Larson/Farber Ch. 4 MEAN, VARIANCE AND STANDARD DEVIATION 12 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:

14. Larson/Farber Ch. 4 MEAN (EXPECTED VALUE) 13 Multiply each value by its probability. Add the products The expected value (the mean) is 1.763 vehicles. Calculate the mean

15. Larson/Farber Ch. 4 CALCULATE THE VARIANCE AND STANDARD DEVIATION 14 The standard deviation is 0.775 vehicles. The mean is 1.763 vehicles. μμμ 2 variance

16. Larson/Farber Ch. 4 EXAMPLE Drawing a Probability Histogram Draw a probability histogram of the following probability distribution which represents the number of DVDs a person rents from a video store during a single visit.

17. Binomial Distributions SECTION 6.2

18. Larson/Farber Ch. 4 17 BINOMIAL EXPERIMENTS 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. Characteristics of a Binomial Experiment The random variable x is a count of the number of successes in n trials.

19. Larson/Farber Ch. 4 18 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

20. Larson/Farber Ch. 4 19 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

21. Larson/Farber Ch. 4 20 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. 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

22. Larson/Farber Ch. 4 21 BINOMIAL PROBABILITIES Find the probability of getting exactly 3 out of 5 questions correct on a quiz that has 4 answer choices for each question. 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 FSSSF 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

23. Larson/Farber Ch. 4 22 COMBINATION OF N VALUES, CHOOSING X 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 There areways.

24. Larson/Farber Ch. 4 23 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

25. Larson/Farber Ch. 4 24 BINOMIAL DISTRIBUTION xP(x) 00.237 10.396 20.264 30.088 40.015 50.001 Binomial Histogram x

26. Larson/Farber Ch. 4 25 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 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

27. Larson/Farber Ch. 4 26 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:

28. More Discrete Probability Distributions SECTION 4.3

29. Larson/Farber Ch. 4 THE GEOMETRIC DISTRIBUTION 28 A marketing study has found that the probability that a person who enters a particular store will make a purchase is 0.30. The probability the first purchase will be made by the first person who enters the store 0.30. That is P(1) = 0.30 The probability the first purchase will be made by the second person who enters the store is (0.70) ( 0.30). So P(2) = (0.70) ( 0.30) = 0.21. The probability the first purchase will be made by the third person who enters the store is (0.70)(0.70)( 0.30). So P(3) = (0.70) (0.70) ( 0.30) = 0.147. The probability the first purchase will be made by person number x is

30. Larson/Farber Ch. 4 29 THE GEOMETRIC DISTRIBUTION A geometric distribution is a discrete probability distribution of the random variable x that satisfies the following conditions. 1.A trial is repeated until a success occurs 2. The repeated trials are independent of each other. 3. The probability of success p is the same for each trial. The probability that the first success will occur on trial number x is where q = 1- p

31. Larson/Farber Ch. 4 30 APPLICATION A cereal maker places a game piece in its boxes. The probability of winning a prize is one in four. Find the probability you a)Win your first prize on the 4 th purchase b) Win your first prize on your 2 nd or 3 rd purchase c) Do not win your first prize in your first 4 purchases. P(4) = (.75) 3. (.25) = 0.1055 P(2) = (.75) 1 (.25) = 0.1875 and P(3) = (.75) 2 (.25) = 0.1406 So P(2 or 3 ) = 0.1875 + 0.1406 = 0.3281 1 – (P(1) + P(2) +P(3) +P(4)) 1 – ( 0.25 + 0.1875 + 0.1406 + 0.1055) = 1 –.6836 = 0.3164

32. Larson/Farber Ch. 4 31 THE POISSON DISTRIBUTION The Poisson distribution is a discrete probability distribution of the random variable x that satisfies the following conditions. 1.The experiment consists of counting the number of times, x, an event occurs in an interval of time, area or space. 2. The probability an event will occur is the same for each interval. 3. The number of occurrences in one interval is independent of the number of occurrences in other intervals. The probability of exactly x occurrences in an interval is e is the irrational number approximately 2.71828  Is the mean number of occurrences per interval.

33. Larson/Farber Ch. 4 APPLICATION It is estimated that sharks kill 10 people each year worldwide. Find the probability a)Three people are killed by sharks this year b) Two or three people are killed by sharks this year 32 P(3)=0.0076 P(2 or 3) = 0.0023 + 0.0076 = 0.0099