1. Chapter 3 Probability Larson/Farber 4th ed

2. Chapter Outline 3.1 Basic Concepts of Probability 3.2 Conditional Probability and the Multiplication Rule 3.3 The Addition Rule 3.4 Additional Topics in Probability and Counting Larson/Farber 4th ed

3. Section 3.1 Basic Concepts of Probability Larson/Farber 4th ed

4. Section 3.1 Objectives Identify the sample space of a probability experiment Identify simple events Use the Fundamental Counting Principle Distinguish among classical probability, empirical probability, and subjective probability Determine the probability of the complement of an event Use a tree diagram and the Fundamental Counting Principle to find probabilities Larson/Farber 4th ed

5. Probability Experiments Probability experiment An action, or trial, through which specific results (counts, measurements, or responses) are obtained. Outcome The result of a single trial in a probability experiment. Sample Space The set of all possible outcomes of a probability experiment. Event Consists of one or more outcomes and is a subset of the sample space. Larson/Farber 4th ed

6. Probability Experiments Probability experiment: Roll a die Outcome: {3} Sample space: {1, 2, 3, 4, 5, 6} Event: {Die is odd}={1, 3, 5} Larson/Farber 4th ed

7. Example: Identifying the Sample Space A probability experiment consists of tossing a three coins. Describe the sample space. Larson/Farber 4th ed Solution: { HHH, HHT, HTT, HTH, HTT, THH, THT, TTH, TTT }

8. Solution: Identifying the Sample Space Larson/Farber 4th ed Tree diagram: The sample space has 8 outcomes: {HHH, HHT, HTT, HTH, THH, THT, TTH, TTT}

9. Simple Events Simple event An event that consists of a single outcome.  e.g. “You randomly select a card from standard deck. Event C is selecting a four of hearts” An event that consists of more than one outcome is not a simple event.  e.g. “A computer is used to randomly select a number between 1 and 200. Event B is selecting a number less than 33.” Larson/Farber 4th ed

10. 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 two events can occur in sequence is m*n. Can be extended for any number of events occurring in sequence. Larson/Farber 4th ed

11. Example: Fundamental Counting Principle Do #14 on page 142. Larson/Farber 4th ed

12. Solution: Fundamental Counting Principle There are three choices of salad, six main dishes, and four desserts. Using the Fundamental Counting Principle: 3 ∙ 6 ∙ 4 = 72 ways Larson/Farber 4th ed

13. Types of Probability Classical (theoretical) Probability Each outcome in a sample space is equally likely. Larson/Farber 4th ed

14. Example: Finding Classical Probabilities 1. Event A: rolling a 3 2. Event B: rolling a 7 3. Event C: rolling a number less than 5 Larson/Farber 4th ed Solution: Sample space: {1, 2, 3, 4, 5, 6} You roll a six-sided die. Find the probability of each event.

15. Solution: Finding Classical Probabilities 1. Event A: rolling a 3 Event A = {3} Larson/Farber 4th ed 2.Event B: rolling a 7 Event B= { } (7 is not in the sample space) 3.Event C: rolling a number less than 5 Event C = {1, 2, 3, 4}

16. Types of Probability Empirical (statistical) Probability Based on observations obtained from probability experiments. Relative frequency of an event. Larson/Farber 4th ed

17. Example: Finding Empirical Probabilities The number of voters (in millions) according to age. Larson/Farber 4th ed Age of Votersf 18 - 205.8 21 - 248.5 25 - 3421.7 35 - 4427.7 45 - 6451.7 65 and older26.7 142.1

18. 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. Larson/Farber 4th ed

19. Types of Probability Subjective Probability Intuition, educated guesses, and estimates. e.g. A doctor may feel a patient has a 90% chance of a full recovery. Larson/Farber 4th ed

20. Range of Probabilities Rule Range of probabilities rule The probability of an event E is between 0 and 1, inclusive. 0 ≤ P(E) ≤ 1 Larson/Farber 4th ed [ ] 00.51 ImpossibleUnlikely Even chance LikelyCertain

21. Complementary Events Complement of event E The set of all outcomes in a sample space that are not included in event E. Denoted E ′ (E prime) P(E ′) + P(E) = 1 P(E) = 1 – P(E ′) P(E ′) = 1 – P(E) Larson/Farber 4th ed E ′ E

22. Example: Probability of the Complement of an Event Back to our voter example: #45 - 48 Larson/Farber 4th ed Age of Votersf 18 - 205.8 21 - 248.5 25 - 3421.7 35 - 4427.7 45 - 6451.7 65 and older26.7 142.1

23. Solution: Probability of the Complement of an Event Use empirical probability to find P(age 25 to 34) = p(E) Larson/Farber 4th ed Use the complement rule, find p( age not 25 to 34) = p(E’) Age of Votersf 18 - 205.8 21 - 248.5 25 - 3421.7 35 - 4427.7 45 - 6451.7 65 and older26.7 142.1

24. Section 3.1 Summary Identified the sample space of a probability experiment Identified simple events Used the Fundamental Counting Principle Distinguished among classical probability, empirical probability, and subjective probability Determined the probability of the complement of an event Used a tree diagram and the Fundamental Counting Principle to find probabilities Larson/Farber 4th ed