1. 1 Definitions Experiment – a process by which an observation ( or measurement ) is observed Sample Space (S)- The set of all possible outcomes (or results) of an experiment Event (E) – a collection of outcomes

2. 2 Example 1) Experiment : Toss a balanced die once and observe its uppermost face Sample Space =S={1,2,3,4,5,6} Events: 1.observe a even number E= { 2,4,6} 2. observe a number less than or equal to 4 F= { 1,2,3,4}

3. 3 Probability Given a event (E), we would like to assign it a number, P(E) P(E) is called the probability of E (likelihood that E will occur) Practical Interpretation The fraction of times that E happens out of a huge number of trials of the same experiment will be close to P(E)

4. 4 Types of Probabilities Theoretical Empirical

5. 5 Theoretical Probabilities Used if the outcomes of an experiment are equally likely to occur If E is an Event

6. 6 Example Toss a balanced die once and observe its uppermost face S={1,2,3,4,5,6} Let G=“observe a number divisible by 3” G={3,6} Then P(G)=2/6=1/3

7. 7 Empirical Probabilities Used when theoretical probabilities cannot be used The experiment is repeated large number of times If E is an Event

8. 8 Example The freshman class at ABC college - 770 students - 485 identified themselves as “smokers” Compute the empirical probability that a randomly selected freshman student from this class is not a smoker

9. 9 Example-contd. E= event that a randomly chosen student from this class is not a smoker P(E)= 285/770=0.37

10. 10 Properties I 1. 2. If E is certain to happen 3. If E and F cannot both happen 4.

11. 11 Union Def. The union of two sets, E and F, is the set of outcomes in E or F and is denoted. Example: E= { 2,4,6} F= { 1,2,3,4}

12. 12 Intersection Def. The intersection of two sets, E and F, is the set of outcomes in E and F and is denoted. Example: E= { 2,4,6} F= { 1,2,3,4}

13. 13 Mutually Exclusive Def. Two events, E and F, are mutually exclusive if they have no outcomes in common, i.e.. If E and F are mutually exclusive, then

14. 14 This property can be extended to more than two events. For any two events, E and F,

15. 15 Complement of an Event Def. The complement of an event, E, is the event that E does not happen and is denoted. Example: S={1,2,3,4,5,6} E= { 2,4,6} Does E and have common outcomes?

16. 16 Since the two events are Mutually Exclusive

17. 17

18. 18 Assign probability to each outcome Each probability must be between 0 and 1 The sum of the probabilities must be equal to 1 If the outcomes of an experiment are all equally likely, then the probability of each outcome is given by,where n is the number of possible outcomes

19. 19 January 26 Reading MBD proj1.ppt (slides 12-23) Topic – Basic Probability

20. 20 DeMorgan’s Laws