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 students 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 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 E or F. 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 E and F. Example: E= { 2,4,6} F= { 1,2,3,4}

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 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

17

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 1. Range 2. Criteria

22 Project Focus – Basic Probability

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