CHAPTER 4 - PROBABILITY
Dec 31, 2015
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