Understanding Probability and Events in Experiments

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. Example 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. 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. Types of Probabilities • Theoretical • Empirical

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

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

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

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

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

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

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

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

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

16. Since the two events are Mutually Exclusive

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. DeMorgan’s Laws

20. Project Focus • How can probability help us with the decision on whether or not to attempt a loan work out? Events: S- an attempted work out is successful F- an attempted work out fails Goal: P(S) – Probability of S or fraction of past work out arrangements which were successful P(F) - Probability of F or fraction of past work out arrangements which were unsuccessful?

21. Using “Countif” function in Excel • Counts the number of cells within a given range that meets the given criteria • Fields for the function • Range • Criteria

22. Project Focus – Basic Probability

23. More on Events S & F • F is the complement of S Recall: