C HAPTER 4 - P ROBABILITY

I NTRODUCTORY V OCABULARY Random (trials) – individual outcomes of a trial are uncertain, but when a large number of trials are performed a regular distribution appears Probability – Proportion of times an outcome would occur in a large number of trials Experimental Probability – What did happen in an experiment. The proportion of times an event occurred in an experiment

Theoretical Probability – What should happen in an experiment. Usually found by looking at experimental probabilities. Probability Models – List of all possible outcomes The probability of each outcome is then listed. Sample Space – the set of all possible outcomes of an event. S = { } Examples: Rolling a die once; S = {1,2,3,4,5,6} Flipping a coin twice; S = {HH,HT,TH,TT}

P ROBABILITY N OTATION A,B,C, etc. – events or outcomes P(A) = the probability of outcome A occuring S = sample space When we represent events, we draw them with Venn Diagrams Venn Diagrams use shapes to represent events and a box around the shapes that represents the sample space or all possible outcomes

G ENERAL S ET T HEORY Union: “or” statements Meaning: joining, addition Symbol: Example 1: Example 2: Set A = {2,4,6,8,10,12} Set B = {1,2,3,4,5,6,7} A B = AB

Intersection: “and” Meaning: overlap, things in common Symbol: Example 1: Example 2: Set A = {2,4,6,8,10,12} Set B = {1,2,3,4,5,6,7} A B = AB

Complement: of event A Meaning: not A. None of the outcomes of event A occur. Everything but A Symbol: A C Example 1: Shade A C Shade A C B Example 2: Set A = {2,4,6,8,10,12} S = {whole numbers 1 to 15} A C = { ABAB

TRY THE SET THEORY WORKSHEET

P ROBABILITY R ULES ! First Three Probability Rules 1. All probabilities lie between 0 and 1 2. Probability of all possible outcomes must be equal to 1 3. Probability of the compliment of A is the same as 1 minus the probability of A Example 1:

Example 2: Example 3: TypeA+A-B+B-AB+AB-O+O- Prob ?

Unions OR => Addition General Rule: Why do we subtract the intersection? We don’t want to count the outcomes in A and B twice, the overlap of A and B. AB

Special Case: What if A and B don’t overlap? So This is called Disjoint or Mutually Exclusive No common outcomes

Conditional Probability Probability of B happening given that A has already happened. Formula: Example: P(A) = 5/10 P(B) = 3/10 P(B|A) = 3/9 since the first one was not replaced P(B|A)=P(A|B)??

Intersections General Rule: Also called the multiplication rule Special Case P(Red) = 3/10 P(Red|Blue) = 3/10 If P(B|A) = P(B) the two events are independent