CHAPTER 4 - PROBABILITY

INTRODUCTORY VOCABULARY

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}

PROBABILITY NOTATION 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

GENERAL SET THEORYUnion: “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 =

A B

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 =

A B

Complement: of event A Meaning: not A. None of the outcomes of

event A occur. Everything but A Symbol: AC

Example 1: Shade AC Shade AC B

Example 2: Set A = {2,4,6,8,10,12}S = {whole numbers 1 to 15}

AC = {

A B

A B

TRY THE SET THEORY WORKSHEET

PROBABILITY RULES!First Three Probability Rules1. All probabilities lie between 0

and 12. Probability of all possible outcomes

must be equal to 13. Probability of the

compliment of A is the same as 1 minus the probability of A

Example 1:

0 ( ) 1P A ( ) 1P S

( ) 1 ( )CP A P A

( )

( )CP H

P H

Example 2:

Example 3:

Type A+ A- B+ B- AB+ AB- O+ O-Prob. 0.16 0.14 0.19 0.17  ? 0.07 0.1 0.11

UnionsOR => AdditionGeneral 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.

( ) ( ) ( ) ( )P A B P A P B P A B

A B

Special Case:What if A and B don’t overlap?

So This is called Disjoint or Mutually ExclusiveNo common outcomes

( ) 0P A B

( ) ( ) ( )P A B P A P B

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)??

( | )P B A

( )( | )

( )

P A BP B A

P A

IntersectionsGeneral Rule:

Also called the multiplication rule

Special CaseP(Red) = 3/10 P(Red|Blue) = 3/10

If P(B|A) = P(B) the two events are independent

( ) ( ) ( | )P A B P A P B A