1. Probability

2. Basic Concepts of Probability What you should learn: How to identify the sample space of a probability experiment and to identify simple events. How to distinguish among classical probability, empirical probability, and subjective probability. How to identify and use properties of probability.

3. Probability Experiment- Vocabulary An experiment through which you obtain counts, measurement or responses. Outcome - The result of a single trial in a probability experiment.

4. Sample Space Vocabulary The set of all possible outcomes of a probability experiment. Event- A subset of the sample space that consists of one or more outcomes of the probability experiment.

5. Identifying Parts of a Probability Experiment. Probability experiment Sample Space Event Outcome Roll a 6-sided die. 1,2,3,4,5,6 Rolling an even number Rolling a 2

6. Simple Event- Vocabulary An event that consists of a single outcome. Example: Rolling a 2 on a die. Non-Example: Selecting an Ace from a standard deck of cards….there are 4 aces.

7. Types of Probability  Classical (Theoretical) Probability  Empirical (Statistical) Probability

8. Classical Probability Vocabulary Used when each outcome in a sample space is equally likely to occur.

9. Probabilities can be expressed as fractions, decimals and percents. FYI This chapter, we will be expressing them as fractions or decimals rounded to three decimal places if necessary. The probability of an impossible event is zero. The probability of an event that is certain to occur is one. 0  P(E)  1 for any event A.

10. Empirical Probability Vocabulary Probability based on observations obtained from probability experiments.

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

12. Types of Probability  Classical (Theoretical) Probability  Empirical (Statistical) Probability  Subjective Probability

13. Subjective Probability Vocabulary Results from intuition, educated guesses and estimates. Example: A company might predict that the chance of employees of the company going on strike is 25%. There is no formula for Subjective Probability.

14. Vocabulary Complement of an Event- The set of all outcomes in a sample space that are not included in event E. Notation E’- read as E prime

15. Properties of Probability P(E) + P(E’) = 1 P(E) = 1 – P(E’) P(E’) = 1 – P(E)

16. Vocabulary Odds - The ratio of the number of successful outcomes to the number of unsuccessful outcomes.

17. Assignment: pg. 111: 1-26,29

18. Conditional Probability and the Multiplication Rule What you should learn: How to find the probability of an event given that another event has occurred. How to distinguish between independent and dependent events. How to use the multiplication rule to find the probability of two events occurring in sequence. How to use the multiplication rule to find conditional probabilities.

19. Conditional Probability Vocabulary The probability of an event occurring, given that another event has already occurred. Notation

20. Example: Two cards are selected in a sequence from a standard deck of 52 cards. Find the probability that the second card is a queen, given that the first card is a king and wasn’t replaced before the second drawing occurred. Examples of Conditional Probability

21. Types of Conditional Probability  Independent Events  Dependent Events

22. Independent Events Vocabulary If the occurrence of one of the events does not affect the probability of the occurrence of the other events. Dependent Events If the occurrence of one of the events affects the probability of the occurrence of the other events.

23. The Multiplication Rule Vocabulary The probability that 2 events A and B will occur in sequence is P(A and B) = P(A) ∙ P(B|A)

24. Assignment: pg. 119: 1,5-10,13-16, 18-20

25. The Addition Rule What you should learn: How to determine if two events are mutually exclusive. How to use the addition rule to find the probability of two events.

26. Mutually Exclusive Vocabulary Two events are mutually exclusive if they can not occur at the same time. A B A and B are mutually exclusive. A B A and B are not mutually exclusive.

27. Eligible voters and 10 year olds. Examples of Mutually Exclusive Events voters 10 yr. olds No overlap

28. Jacks in a deck and threes in a deck Examples of Mutually Exclusive Events jacks threes No overlap

29. Spinning a spinner and rolling a dice. Examples of Mutually Exclusive Events spins numbers No overlap

30. Jacks and diamonds Examples of Non-Mutually Exclusive Events jacks diamonds overlap jj

31. Sophomores and boys Examples of Non-Mutually Exclusive Events 10 th graders boys overlap 10 th gr. boys

32. The Addition Rule The probability that events A or B will occur is given by… Where P(A and B) would be the overlap section.

33. Assignment: pg. 129: 2-18

34. Counting Principles What you should learn: How to use the Fundamental Counting Principle to find the number of ways two or more events can occur. How to find the number of ways a group of objects can be arranged in order. How to find the number of ways to choose several objects from a group without regard to order. How to use counting principles to find probabilities.

35. The Fundamental Counting Principle - Vocabulary If one event can occur in m ways, and a second event can occur in n ways, the number of ways the 2 events can occur in sequence is m∙n. This rule can be extended for any number of events occurring in sequence.

36. Vocabulary Factorial - A multiplication pattern denoted by n!. It is the product of n with each of the positive counting numbers less than n. n! = n(n-1)(n-2)(n-3)…(1) Special Definition: 0! = 1

37. Permutation - Vocabulary An ordered arrangement of objects. The number of permutations of n distinct objects is n!.

38. Example of a Permutation How many permutations are their for 3 people to fill 3 vacant positions at Corporation Z? There would be 3! arrangements to fill these vacancies. 3! = 3∙2∙1 = 6 arrangements A,B,C A,C,B B,A,C B,C,A C,A,B C,B,A

39. Permutation of n items taken r at a time- What if I don’t want to use all the items. Keep in mind….Order is Important.

40. Permutation of n items taken r at a time with duplicates- Distinguishable Permutations Keep in mind….Order is still Important.

41. Combination - Vocabulary An arrangement of objects where order does not matter. Suppose you want to buy 3 CDs from a selection of 5?

42. Example of a Combination The tree diagram would lead to the following outcomes. ABC ABD ABE ACB ACD ACE ADB ADC ADE AEB AEC AED BAC BAD BAE BCA BCD BCE BDA BDC BDE BEA BEC BED CAB CAD CAE CBA CBD CBE CDA CDB CDE CEA CEB CED DAB DAC DAE DBA DBC DBE DCA DCB DCE DEA DEB DEC EAB EAC EAD EBA EBC EBD ECA ECB ECD EDA EDB EDC Now take care of duplicates… There are 10 combinations.

43. There has to be an easier way!!!

44. Combinations

45. Assignment: pg. 140: 1-28

46. Application of the Counting Principles You can determine probabilities if your can determine how many ways a particular event can occur.

47. Assignment: pg. 142: 29-30