1. © 2005 McGraw-Hill Ryerson Ltd. 4-1 Statistics A First Course Donald H. Sanders Robert K. Smidt Aminmohamed Adatia Glenn A. Larson

2. © 2005 McGraw-Hill Ryerson Ltd. 4-2 Chapter 4 Probability Concepts

3. © 2005 McGraw-Hill Ryerson Ltd. 4-3 Chapter 4 - Topics Some Basic Considerations Probabilities for Compound Events Random Variables, Probability Distributions, and Expected Values

4. © 2005 McGraw-Hill Ryerson Ltd. 4-4 Some Basic Considerations Probability Experiment –Any action for which an outcome, response, or measurement if obtained that can’t be predicted with certainty Event –Subset or collection of outcomes from the sample space Simple Event –An event that can’t be broken down any further Sample Space –Set of all possible simple outcomes, responses, or measurement of an experiment

5. © 2005 McGraw-Hill Ryerson Ltd. 4-5 Some Basic Considerations Classical (a priori) Probability Relative Frequency (Empirical) Probability Probability –Relative likelihood of that event occurring

6. © 2005 McGraw-Hill Ryerson Ltd. 4-6 Probabilities for Compound Events Compound Event –Combination of two or more events Conditional Probability –Probability that one event will occur given that another event has already occurred Joint Probability –Probability that two or more events occur such that the second event occurs after the first event Events could be dependent or independent

7. © 2005 McGraw-Hill Ryerson Ltd. 4-7 A tree diagram showing all the possible outcomes of drawing 2 pieces from 5 red, 3 green, and 2 yellow candies. Note that the total probability is 90/90 or 1. Figure 4.5

8. © 2005 McGraw-Hill Ryerson Ltd. 4-8

9. © 2005 McGraw-Hill Ryerson Ltd. 4-9 Probabilities for Compound Events Joint Probability: Multiplication Rule Dependent Events –The occurrence of one event affects the probability of the occurrence of another event where P(B|A) is the conditional probability of event B occurring given that event A has occurred

10. © 2005 McGraw-Hill Ryerson Ltd. 4-10 Probabilities for Compound Events Joint Probability: Multiplication Rule Independent Events –The occurrence of one event does not affect the probability of the occurrence of another event

11. © 2005 McGraw-Hill Ryerson Ltd. 4-11 Probabilities for Compound Events Mutually Exclusive Events –Events cannot occur at the same time Addition Rule Venn Diagram

12. © 2005 McGraw-Hill Ryerson Ltd. 4-12 Probabilities for Compound Events Non-mutually Exclusive Events –Events can occur at the same time Addition Rule Venn Diagram

13. © 2005 McGraw-Hill Ryerson Ltd. 4-13 Probabilities for Compound Events Complement of an Event –Consists of all possible outcomes from the sample space that are not in event Rule for Complementary Events

14. © 2005 McGraw-Hill Ryerson Ltd. 4-14 Random Variables, Probability Distributions, and Expected Values Random Variable –Single numerical value for each outcome of a probability experiment Discrete Random Variable –All possible values can be counted or listed Continuous Random Variable –Infinite number of values that can fall, without interruption, along an unbroken interval

15. © 2005 McGraw-Hill Ryerson Ltd. 4-15 Probability Distribution –Discrete random variable –Gives probability for each of the values of the random variable Random Variables, Probability Distributions, and Expected Values

16. © 2005 McGraw-Hill Ryerson Ltd. 4-16 Distribution of the sum of the numbers thrown on one toss of two fair dice Figure 4.6

17. © 2005 McGraw-Hill Ryerson Ltd. 4-17

18. © 2005 McGraw-Hill Ryerson Ltd. 4-18 Figure 4.6 A histogram of the probability distribution of the results of throwing a pair of dice.

19. © 2005 McGraw-Hill Ryerson Ltd. 4-19

20. © 2005 McGraw-Hill Ryerson Ltd. 4-20 Expected Values: Mean of Random Variable Formula Random Variables, Probability Distributions, and Expected Values Standard Deviation of Random Variable Formula Method 1 Shortcut Method

21. © 2005 McGraw-Hill Ryerson Ltd. 4-21 End of Chapter 4 Probability Concepts