1. 5A-1

2. Probability (Part 1) Random Experiments Random Experiments Probability Rules of Probability Rules of Probability Independent Events Independent Events Chapter 5A5A McGraw-Hill/Irwin© 2008 The McGraw-Hill Companies, Inc. All rights reserved.

3. 5A-3 A random experiment is an observational process whose results cannot be known in advance.A random experiment is an observational process whose results cannot be known in advance. The set of all outcomes (S) is the sample space for the experiment.The set of all outcomes (S) is the sample space for the experiment. A sample space with a countable number of outcomes is discrete.A sample space with a countable number of outcomes is discrete.  Sample Space Random Experiments

4. 5A-4 For example, when CitiBank makes a consumer loan, the sample space is:For example, when CitiBank makes a consumer loan, the sample space is: S = {default, no default} The sample space describing a Wal-Mart customer’s payment method is:The sample space describing a Wal-Mart customer’s payment method is: S = {cash, debit card, credit card, check}  Sample Space Random Experiments

5. 5A-5 For a single roll of a die, the sample space is:For a single roll of a die, the sample space is: S = {1, 2, 3, 4, 5, 6} When two dice are rolled, the sample space is the following pairs:When two dice are rolled, the sample space is the following pairs:  Sample Space Random Experiments {(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)} S =S =S =S =

6. 5A-6 An event is any subset of outcomes in the sample space.An event is any subset of outcomes in the sample space. A simple event or elementary event, is a single outcome.A simple event or elementary event, is a single outcome. A discrete sample space S consists of all the simple events (E i ):A discrete sample space S consists of all the simple events (E i ): S = {E 1, E 2, …, E n }  Events Random Experiments

7. 5A-7 What are the chances of observing a H or T?What are the chances of observing a H or T? These two elementary events are equally likely.These two elementary events are equally likely. S = {H, T} Consider the random experiment of tossing a balanced coin. What is the sample space?Consider the random experiment of tossing a balanced coin. What is the sample space? When you buy a lottery ticket, the sample space S = {win, lose} has only two events.When you buy a lottery ticket, the sample space S = {win, lose} has only two events.  Events Random Experiments Are these two events equally likely to occur?Are these two events equally likely to occur?

8. 5A-8 For example, in a sample space of 6 simple events, we could define the compound eventsFor example, in a sample space of 6 simple events, we could define the compound events A compound event consists of two or more simple events.A compound event consists of two or more simple events. These are displayed in a Venn diagram:These are displayed in a Venn diagram: A = {E 1, E 2 } B = {E 3, E 5, E 6 }  Events Random Experiments

9. 5A-9 Many different compound events could be defined. Compound events can be described by a rule. S = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} For example, the compound event A = “rolling a seven” on a roll of two dice consists of 6 simple events:  Events Random Experiments

10. 5A-10 The probability of an event is a number that measures the relative likelihood that the event will occur.The probability of an event is a number that measures the relative likelihood that the event will occur. The probability of event A [denoted P(A)], must lie within the interval from 0 to 1:The probability of event A [denoted P(A)], must lie within the interval from 0 to 1: 0 < P(A) < 1 If P(A) = 0, then the event cannot occur. If P(A) = 1, then the event is certain to occur.  Definitions ProbabilityProbability

11. 5A-11 In a discrete sample space, the probabilities of all simple events must sum to unity:In a discrete sample space, the probabilities of all simple events must sum to unity: For example, if the following number of purchases were made byFor example, if the following number of purchases were made by P(S) = P(E 1 ) + P(E 2 ) + … + P(E n ) = 1 credit card: 32% 32% debit card: 20% 20% cash: 35% 35% check: 18% 18% Sum = 100%  Definitions ProbabilityProbability P(credit card) =.32 P(debit card) =.20 P(cash) =.35 P(check) =.18 Sum = 1.0 Probability

12. 5A-12 The law of large numbers is an important probability theorem that states that a large sample is preferred to a small one.The law of large numbers is an important probability theorem that states that a large sample is preferred to a small one. Flip a coin 50 times. We would expect the proportion of heads to be near.50.Flip a coin 50 times. We would expect the proportion of heads to be near.50. A large n may be needed to get close to.50.A large n may be needed to get close to.50. However, in a small finite sample, any ratio can be obtained (e.g., 1/3, 7/13, 10/22, 28/50, etc.).However, in a small finite sample, any ratio can be obtained (e.g., 1/3, 7/13, 10/22, 28/50, etc.).  Law of Large Numbers ProbabilityProbability Consider the results of 10, 20, 50, and 500 coin flips.Consider the results of 10, 20, 50, and 500 coin flips.

13. 5A-13 ProbabilityProbability

14. 5A-14 The complement of an event A is denoted by A′ and consists of everything in the sample space S except event A.The complement of an event A is denoted by A′ and consists of everything in the sample space S except event A.  Complement of an Event Rules of Probability

15. 5A-15 Since A and A′ together comprise the entire sample space, P(A) + P(A′ ) = 1Since A and A′ together comprise the entire sample space, P(A) + P(A′ ) = 1 The probability of A′ is found by P(A′ ) = 1 – P(A)The probability of A′ is found by P(A′ ) = 1 – P(A) For example, The Wall Street Journal reports that about 33% of all new small businesses fail within the first 2 years. The probability that a new small business will survive is:For example, The Wall Street Journal reports that about 33% of all new small businesses fail within the first 2 years. The probability that a new small business will survive is: P(survival) = 1 – P(failure) = 1 –.33 =.67 or 67%  Complement of an Event Rules of Probability

16. 5A-16 The odds in favor of event A occurring isThe odds in favor of event A occurring is Odds are used in sports and games of chance.Odds are used in sports and games of chance. For a pair of fair dice, P(7) = 6/36 (or 1/6). What are the odds in favor of rolling a 7?For a pair of fair dice, P(7) = 6/36 (or 1/6). What are the odds in favor of rolling a 7?  Odds of an Event Rules of Probability

17. 5A-17 On the average, for every time a 7 is rolled, there will be 5 times that it is not rolled.On the average, for every time a 7 is rolled, there will be 5 times that it is not rolled. In other words, the odds are 1 to 5 in favor of rolling a 7.In other words, the odds are 1 to 5 in favor of rolling a 7. The odds are 5 to 1 against rolling a 7.The odds are 5 to 1 against rolling a 7.  Odds of an Event Rules of Probability In horse racing and other sports, odds are usually quoted against winning.In horse racing and other sports, odds are usually quoted against winning.

18. 5A-18 If the odds against event A are quoted as b to a, then the implied probability of event A is:If the odds against event A are quoted as b to a, then the implied probability of event A is: For example, if a race horse has a 4 to 1 odds against winning, the P(win) isFor example, if a race horse has a 4 to 1 odds against winning, the P(win) is P(A) =  Odds of an Event Rules of Probability P(win) = or 20%

19. 5A-19 The union of two events consists of all outcomes in the sample space S that are contained either in event A or in event B or both (denoted A  B or “A or B”).  may be read as “or” since one or the other or both events may occur.  Union of Two Events Rules of Probability

20. 5A-20 For example, randomly choose a card from a deck of 52 playing cards.For example, randomly choose a card from a deck of 52 playing cards. It is the possibility of drawing either a queen (4 ways) or a red card (26 ways) or both (2 ways).It is the possibility of drawing either a queen (4 ways) or a red card (26 ways) or both (2 ways). If Q is the event that we draw a queen and R is the event that we draw a red card, what is Q  R?If Q is the event that we draw a queen and R is the event that we draw a red card, what is Q  R?  Union of Two Events Rules of Probability

21. 5A-21 The intersection of two events A and B (denoted A  B or “A and B”) is the event consisting of all outcomes in the sample space S that are contained in both event A and event B.The intersection of two events A and B (denoted A  B or “A and B”) is the event consisting of all outcomes in the sample space S that are contained in both event A and event B.  may be read as “and” since both events occur. This is a joint probability.  Intersection of Two Events Rules of Probability

22. 5A-22 It is the possibility of getting both a queen and a red card (2 ways).It is the possibility of getting both a queen and a red card (2 ways). If Q is the event that we draw a queen and R is the event that we draw a red card, what is Q  R?If Q is the event that we draw a queen and R is the event that we draw a red card, what is Q  R? For example, randomly choose a card from a deck of 52 playing cards.For example, randomly choose a card from a deck of 52 playing cards.  Intersection of Two Events Rules of Probability

23. 5A-23 The general law of addition states that the probability of the union of two events A and B is:The general law of addition states that the probability of the union of two events A and B is: P(A  B) = P(A) + P(B) – P(A  B) When you add the P(A) and P(B) together, you count the P(A and B) twice. So, you have to subtract P(A  B) to avoid over- stating the probability. A B A and B  General Law of Addition Rules of Probability

24. 5A-24 For the card example:For the card example: P(Q) = 4/52 (4 queens in a deck) = 4/52 + 26/52 – 2/52 P(Q  R) = P(Q) + P(R) – P(Q  Q) Q 4/52 R 26/52 Q and R = 2/52  General Law of Addition Rules of Probability = 28/52 =.5385 or 53.85% P(R) = 26/52 (26 red cards in a deck) P(Q  R) = 2/52 (2 red queens in a deck)

25. 5A-25 Events A and B are mutually exclusive (or disjoint) if their intersection is the null set (  ) that contains no elements.Events A and B are mutually exclusive (or disjoint) if their intersection is the null set (  ) that contains no elements. If A  B = , then P(A  B) = 0 In the case of mutually exclusive events, the addition law reduces to:In the case of mutually exclusive events, the addition law reduces to: P(A  B) = P(A) + P(B)  Mutually Exclusive Events Rules of Probability  Special Law of Addition

26. 5A-26 The probability of event A given that event B has occurred.The probability of event A given that event B has occurred. Denoted P(A | B). The vertical line “ | ” is read as “given.”Denoted P(A | B). The vertical line “ | ” is read as “given.” for P(B) > 0 and undefined otherwise  Conditional Probability Rules of Probability

27. 5A-27 Consider the logic of this formula by looking at the Venn diagram. The sample space is restricted to B, an event that has occurred. A  B is the part of B that is also in A. The ratio of the relative size of A  B to B is P(A | B).  Conditional Probability Rules of Probability

28. 5A-28 Of the population aged 16 – 21 and not in college:Of the population aged 16 – 21 and not in college: Unemployed13.5% High school dropouts 29.05% Unemployed high school dropouts 5.32% 5.32% What is the conditional probability that a member of this population is unemployed, given that the person is a high school dropout?What is the conditional probability that a member of this population is unemployed, given that the person is a high school dropout?  Example: High School Dropouts Rules of Probability

29. 5A-29 First defineFirst define U = the event that the person is unemployed D = the event that the person is a high school dropout P(U) =.1350 P(D) =.2905 P(U  D) =.0532 or 18.31% or 18.31% P(U | D) =.1831 > P(U) =.1350P(U | D) =.1831 > P(U) =.1350 Therefore, being a high school dropout is related to being unemployed.Therefore, being a high school dropout is related to being unemployed.  Example: High School Dropouts Rules of Probability

30. 5A-30 Event A is independent of event B if the conditional probability P(A | B) is the same as the marginal probability P(A).Event A is independent of event B if the conditional probability P(A | B) is the same as the marginal probability P(A). To check for independence, apply this test:To check for independence, apply this test: If P(A | B) = P(A) then event A is independent of B. Another way to check for independence:Another way to check for independence: If P(A  B) = P(A)P(B) then event A is independent of event B since P(A | B) = P(A  B) = P(A)P(B) = P(A) P(B) P(B) P(B) P(B) Independent Events

31. 5A-31 Out of a target audience of 2,000,000, ad A reaches 500,000 viewers, B reaches 300,000 viewers and both ads reach 100,000 viewers.Out of a target audience of 2,000,000, ad A reaches 500,000 viewers, B reaches 300,000 viewers and both ads reach 100,000 viewers. What is P(A | B)?What is P(A | B)? Independent Events  Example: Television Ads.3333 or 33%.3333 or 33%

32. 5A-32 So, P(ad A) =.25 P(ad B) =.15 P(A  B) =.05 P(A | B) =.3333So, P(ad A) =.25 P(ad B) =.15 P(A  B) =.05 P(A | B) =.3333 P(A | B) =.3333 ≠ P(A) =.25P(A | B) =.3333 ≠ P(A) =.25 P(A)P(B)=(.25)(.15)=.0375 ≠ P(A  B)=.05P(A)P(B)=(.25)(.15)=.0375 ≠ P(A  B)=.05 Are events A and B independent?Are events A and B independent? Independent Events  Example: Television Ads

33. 5A-33 When P(A) ≠ P(A | B), then events A and B are dependent.When P(A) ≠ P(A | B), then events A and B are dependent. For dependent events, knowing that event B has occurred will affect the probability that event A will occur.For dependent events, knowing that event B has occurred will affect the probability that event A will occur. For example, knowing a person’s age would affect the probability that the individual uses text messaging but causation would have to be proven in other ways.For example, knowing a person’s age would affect the probability that the individual uses text messaging but causation would have to be proven in other ways. Independent Events  Dependent Events Statistical dependence does not prove causality.Statistical dependence does not prove causality.