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Chapter 3

Chapter 3. Probability. Section 3.1. 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.

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Chapter 3

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  1. Chapter 3 Probability

  2. Section 3.1 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. Vocabulary Probability Experiment- An experiment through which you obtain counts, measurement or responses. Outcome - The result of a single trial in a probability experiment.

  4. Vocabulary Sample Space 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 Roll a 6-sided die. • Sample Space 1,2,3,4,5,6 • Event Rolling an even number • Outcome Rolling a 2

  6. Vocabulary Simple Event- 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. Vocabulary Classical Probability Used when each outcome in a sample space is equally likely to occur.

  9. FYI Probabilities can be expressed as fractions, decimals and percents. 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. Vocabulary Empirical Probability 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. Vocabulary Subjective Probability 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. Section 3.2 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. Vocabulary Conditional Probability The probability of an event occurring, given that another event has already occurred. Notation

  20. Examples of Conditional Probability 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.

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

  22. Vocabulary Independent Events 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. Vocabulary The Multiplication Rule 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. Section 3.3 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. Vocabulary Mutually Exclusive Two events are mutually exclusive if they can not occur at the same time. B A B A A and B are mutually exclusive. A and B are not mutually exclusive.

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

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

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

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

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

  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. Section 3.4 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. Vocabulary The Fundamental Counting Principle - 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. Vocabulary Permutation - 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 C,B,A A,C,B B,C,A B,A,C C,A,B A,B,C

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

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

  41. Vocabulary Combination - 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 Now take care of duplicates… Suppose you want to buy 3 CDs from a selection of 5? There are 10 combinations. The tree diagram would lead to the following outcomes. CBA CBD CBE CDA CDB CDE BCA BCD BCE CAB CAD CAE CEA CEB CED ABC ABD ABE AEB AEC AED BDA BDC BDE ACB ACD ACE BAC BAD BAE BEA BEC BED ADB ADC ADE ECA ECB ECD EAB EAC EAD EDA EDB EDC DCA DCB DCE EBA EBC EBD DAB DAC DAE DEA DEB DEC DBA DBC DBE

  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

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