Probability

1. Probability

2. Important Terms Probability experiment: An action through which counts, measurements or responses are obtained Sample space: The set of all possible outcomes Event: A subset of the sample space. Outcome: The result of a single trial

3. Example Probability experiment: Roll a die Sample space: { 1 2 3 4 5 6 } Event: { Die is even } = { 2 4 6 } Outcome: {4}

4. Practice • Use a tree diagram to develop the sample space that results from rolling two six-sided dice.

5. Tree Diagrams Two dice are rolled. Describe the sample space. Start 1st roll 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 2nd roll 36 outcomes

6. Sample Space and Probabilities Two dice are rolled and the sum is noted. 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 Find the probability the sum is 4. 3/36 = 1/12 = 0.083 Find the probability the sum is 11. 2/36 = 1/18 = 0.056 Find the probability the sum is 4 or 11. 5/36 = 0.139

7. Ice Cream Sundaes Consider an ice cream shop with 31 flavors of ice cream and 15 different toppings. • How many different sundaes can you make if you use 1 scoop of ice cream and two different toppings? • What if you use 3 different scoops and 2 toppings?

8. More multiplication principle • How many different outcomes can you have if you flip a coin 3 times? • How many different outcomes can you have if you flip a coin and roll a die?

9. Theoretical P(A) = number if ways A can occur total number of outcomes In a bag you have 3 red marbles, 2 blue marbles and 7 yellow marbles. If you select one marble at random, P(red) = 3 / (3+2+7) = 3/12 = 1/4

10. Theoretical Examples • Using a standard deck of cards, find the probability of the following. • Selecting a seven • Selecting a diamond • Selecting a diamond, heart, club or spade • Selecting a face card

11. Empirical Probability is based on observations or experiments.

12. Empirical examples • A pond contains three types of fish: bluegills, redgills and crappies. You catch 40 fish and record the type. The following frequency distribution shows your results. • Fish Type Number of times caught • Bluegill 13 • Redgill 17 • Crappy 10 • If you catch a fish, what is the probability that it is a redgill?

13. Empirical examples • What is the probability of getting a bluegill? • What is the probability of not getting a crappy?

14. Subjective • Subjective probability results from educated guesses, intuition and estimates. • A doctor’s prediction that a patient has a 90% chance of full recovery • A business analyst predicting an employee strike being 0.25

15. Summary • Classical (Theoretical) • The number of outcomes in a sample space is known and each outcome is equally likely to occur. • Empirical (Statistical) • The frequency of outcomes in the sample space is estimated from experimentation. • Subjective (Intuition) • Probabilities result from intuition, educated guesses, and estimates.

16. Probability • If P(E) = 0, then event E is impossible. • If P(E) = 1, then event E is certain. • 0  P(E)  1 • Impossible Even Certain • 0 .5 1

17. E Complementary Events The complement of event E is event E´. E´ consists of all the events in the sample space that are not in event E. P(E´) = 1 - P(E) E´

18. Example The day’s production consists of 12 cars, 5 of which are defective. If one car is selected at random, find the probability it is not defective. Solution: P(defective) = 5/12 P(not defective) = 1 - 5/12 = 7/12 = 0.583

19. Examples • What is the probability that a family with 3 children does not have 2 boys and 1 girl? • What is the probability that you do not get a pair of sixes when you roll 2 dice?

20. 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.

21. Example A company tracks the number of sales new employees make each day during a 100-day probationary period. The results for one new employee are shown below. Construct and graph the probability distribution. Sales per day, x 0 1 2 3 4 5 6 7 number of days, f 16 19 15 21 9 10 8 2

22. Example Decide whether each distribution is a probability distribution. Explain your reasoning. x 5 6 7 8 P(x) 1/16 5/8 ¼ 3/16

23. Example Decide whether each distribution is a probability distribution. Explain your reasoning. x 1 2 3 4 P(x) 0.09 0.36 0.49 0.06

24. Odds • Odds for an event A are P(A) P(not A) In sports we often look at wins over losses.

25. Independent Events • Two events are independent if the occurrence (or non-occurrence) of one of the events does not affect the probability of the occurrence of the other event. • Events that are not independent are dependent.

26. Examples Independent Events A = Being female B = Having type O blood A = 1st child is a boy B = 2nd child is a boy Dependent Events A = taking an aspirin each day B = having a heart attack A = being a female B = being under 64” tall

27. Examples • Determine if the following events are independent or dependent. • 1. 12 cars are on a production line where 5 are defective and 2 cars are selected at random. • A = first car is defective • B = second car is defective. • Two dice are rolled. • A = first is a 4 and B = second is a 4

28. Multiplication Rule To find the probability that two events, A and B will occur in sequence, multiply the probability A occurs by the conditional probability B occurs, given A has occurred. P(A and B) = P(A) x P(B given A) Two cars are selected from a production line of 12 where 5 are defective. Find the probability both cars are defective. A = first car is defective B = second car is defective. P(A) = 5/12 P(B given A) = 4/11 P(A and B) = 5/12 x 4/11 = 5/33 = 0.1515

29. Multiplication Rule Two dice are rolled. Find the probability both are 4’s. A = first die is a 4 and B = second die is a 4.

30. Independent Events • When two events A and B are independent, then P (A and B) = P(A) x P(B) • ***Note for independent events P(B) and P(B given A) are the same.

31. B B A A Compare “A and B” to “A or B” The compound event “A and B” means that A and B both occur in the same trial. Use the multiplication rule to find P(A and B). The compound event “A or B” means either A can occur without B, B can occur without A or both A and B can occur. Use the addition rule to find P(A or B). A or B A and B

32. Mutually Exclusive Events Two events, A and B, are mutually exclusive if they cannot occur in the same trial. A = A person is under 21 years old B = A person is running for the U.S. Senate A = A person was born in Philadelphia B = A person was born in Houston A Mutually exclusive B P(A and B) = 0 When event A occurs it excludes event B in the same trial.

33. B A Non-mutually exclusive P(A and B) ≠ 0 Non-Mutually Exclusive Events If two events can occur in the same trial, they are non-mutually exclusive. A = A person is under 25 years old B = A person is a lawyer A = A person was born in Philadelphia B = A person watches West Wing on TV A and B

34. Examples • Determine whether the events are mutually exclusive or not. • Roll a die • A: Roll a 3 B: Roll a 4 • Select a student • A: select a male student B: select a nursing major • Select a blood donor • A: donor is type O B: donor is female

35. Examples • Select a card from a standard deck • A: the card is a jack • B: the card is a face card • Select a student • A: the student is 16 years old • B: the student has blue eyes • Select a registered vehicle • A: the vehicle is a Ford • B: the vehicle is a Toyota

36. The Addition Rule The probability that one or the other of two events will occur is: P(A) + P(B) – P(A and B) A card is drawn from a deck. Find the probability it is a king or it is red. A = the card is a king B = the card is red. P(A) = 4/52 and P(B) = 26/52 butP(A and B) = 2/52 P(A or B) = 4/52 + 26/52– 2/52 = 28/52 = 0.538

37. The Addition Rule A card is drawn from a deck. Find the probability the card is a king or a 10. A = the card is a king B = the card is a 10. P(A) = 4/52 and P(B) = 4/52 and P(A and B) = 0/52 P(A or B) = 4/52 + 4/52 – 0/52 = 8/52 = 0.054 When events are mutually exclusive, P(A or B) = P(A) + P(B)

38. Examples 1. A die is rolled. Find the probability of rolling a 6 or an odd number. -are the events mutually exclusive? -find P(A), P(B) and, if necessary, P(A and B) -use the addition rule to find the probability

39. Example 2. A card is selected from a standard deck. Find the probability that the card is a face card or a a heart. -are the events mutually exclusive? -find P(A), P(B) and, if necessary, P(A and B) -use the addition rule to find the probability

40. Contingency Table The results of responses when a sample of adults in 3 cities was asked if they liked a new juice is: Omaha Seattle Miami Total Yes 100 150 150 400 No 125 130 95 350 Undecided 75 170 5 250 Total 300 450 250 1000 One of the responses is selected at random. Find: 3. P(Miami or Yes) 4. P(Miami or Seattle) 1. P(Miami and Yes) 2. P(Miami and Seattle)

41. Contingency Table Omaha Seattle Miami Total Yes 100 150 150 400 No 125 130 95 350 Undecided 75 170 5 250 Total 300 450 250 1000 One of the responses is selected at random. Find: 1. P(Miami and Yes) 2. P(Miami and Seattle) = 250/1000 • 150/250 = 150/1000 = 0.15 =0

42. Contingency Table Omaha Seattle Miami Total Yes 100 150 150 400 No 125 130 95 350 Undecided 75 170 5 250 Total 300 450 250 1000 3 P(Miami or Yes) 4. P(Miami or Seattle) 250/1000 + 400/1000 – 150/1000 = 500/1000 = 0.5 250/1000 + 450/1000 – 0/1000 = 700/1000 = 0.7

43. Summary For complementary events P(E') = 1 - P(E) Subtract the probability of the event from one. The probabilityboth of two events occur P(A and B) = P(A) • P(B given A) Multiply the probability of the first event by the conditional probability the second event occurs, given the first occurred.

44. Summary Probability at least one of two events occur P(A or B) = P(A) + P(B) - P(A and B) Add the simple probabilities, but to prevent double counting, don’t forget to subtract the probability of both occurring.

45. Law of Large Numbers • As an experiment is repeated over and over, the empirical probability of an event approaches the theoretical (actual) probability of the event.

46. Expected Value • Average- what you would expect given a probability distribution. • To find the expected value, multiply each event value by its probability and add

47. Example X P(X) 0 .25 1 .50 2 .15 3 .10 0(.25) + 1(.5) + 2(.15) + 3(.10) = 1.1

48. Example A bus arrives at a bus stop at noon, 12:20 and 1:00. You arrive at the bus stop at random times between noon and 1:00 every day so all arrival times are equally likely. a. What is the probability that you will arrive at the bus stop between noon and 12:20? What is the mean wait time in that case?

49. What is the probability that you will arrive at the bus stop between 12:20 and 1:00? What is your mean wait time in that case?

50. Overall, what is your expected waiting time for the bus? • Would you expect your waiting time to be longer or shorter if the bus arrived at equally spaced intervals (say noon, 12:30 and 1:00)?