1. Probability Notes
Math 309

2. Some Definitions Experiment - means of making an observation
Sample Space (S) - set of all outcomes of an experiment listed in a mutually exclusive and exhaustive manner
Event - subset of a sample space
Simple Event - an event which can only happen in one way; (or can be thought of as a sample point - a one element subset of S)

3. Since events are sets, we need to understand the basic set operations
Intersection everything in A and B
Union everything in A or B or both
Complement everything not in A

4. You should be able to sketch Venn diagrams to describe the intersections, unions, & complements of sets.
Note that these set operations obey the commutative, associative, and distributive laws

5. DeMorgan’s Laws
Convince yourself that these are reasonable with Venn diagrams!

6. Another definition -
A and B are mutually exclusive iff
A  B = 

7. Axioms of Probability (these are FACT, no proof needed!)
Let E represent an event, S the sample space,
For pairwise mutually exclusive events, the probability of their union is the sum of their respective probabilities, i.e.

8. Theorems (You should be able to prove these using the axioms and definitions.)
Thm The probability of the empty set is zero.
Thm 1.2 – Let {A1, A2, ,An} be a mutually exclusive set of events. Then
P(A1A2 An) = P(A1) + P(A2) P(An)

9. Propositions (You should be able to prove these using the axioms and definitions.)
Let E and F be any two events.
P(E) 1
If E is a subset of F, then P(E) P(F).
For mutually exclusive events, P(E F) = P(E) + P(F)

10. Theorems (You should be able to prove these using the axioms and definitions.)
Let E and F be any two events.
Thm P( ) = 1 - P(E)
Thm P(E F) = P(E) + P(F) - P(E F)

11. Unions get complicated if events are not mutually exclusive!
P(A  B  C) = P(A) + P(B) + P(C)
- P(A  B) - P(A  C) - P(B  C)
+ P(A  B  C) B

12. Let A and B be any two events.
More Theorems
Let A and B be any two events.
Thm 1.5- If A  B,
then P(B-A) = P(B AC) = P(B) – P(A)
Corollary: If A is a subset of B, then P(A) <= P(B).
Thm P(A) = P(A  B) + P(A  BC) (generalize)

13. Sample Spaces with Equally Likely Outcomes
In an experiment where all simple events (sample points) are equally likely, one can find the probability of an event by counting two sets.

14. Combinatorial Methods
Math 309

15. Combinatorics Basic Principle of Counting Permutations Combinations
(a.k.a. Multiplication Principle)
Permutations
Permutations with indistinguishable objects
Combinations

16. Basic Counting Principle
If experiment 1 has m outcomes and experiment 2 has n outcomes, then there are m*n outcomes for both experiments.
The principle can be generalized for r experiments. The number of outcomes of r experiments is the product of the number of outcomes of each experiment.

17. We define experiment as a means of making an observation (e. g
We define experiment as a means of making an observation (e.g. flip a coin, choose a color).
Each experiment could be making a choice from a different set.

18. Permutations # of arrangements of one set, order matters
application of the basic counting principle where we return to the same set for the next selection
P(n,r) = n!/(n-r)!

19. Permutations with Indistinguishable Objects
Order the objects as if they were distinguishable
Then “divide out” those arrangements that look identical.

20. Combinations the number of selections, order doesn’t matter
C(n,r) = n!/[(n-r)!r!]
the number of arrangements can be counted by selecting the objects and then ordering them
i.e. P(n,r) = C(n,r)*r!

21. Observations about Combinations
C(n, r) = C(n, n-r)
C(n, n) = C(n, 0) = 1
C(n, 1) = n = C(n, n-1)
C(n, 2) = n(n-1)/2

22. Combining Counting Techniques
If we are careful with language,
when we say “AND”, we multiply
“AND”  multiplication  intersection
when we say “OR”, we add
“OR”  addition  union

23. Conditional Probability and Independence

24. Conditional Probability P(A|B)
P(A|B) is read,
“the probability of A given B”
B is known to occur.

25. The multiplication rule and  intersectionmultiply
P(A  B) = P(A)*P(B|A)
= P(B)*P(A|B)
Intersections get more complicated when there are more events, e.g.
P(ABCD) =
P(A)* P(B|A)*P(C|AB)*P(D|A BC)

26. Independent Events
A and B are independent if any of the following are true:
P(AB) = P(A)*P(B)
P(A|B) = P(A)
P(B|A) = P(B)
You need to check probabilities to determine if events are independent.
If A, B, C, & D are pairwise independent,
P (AB C D) = P(A)*P(B)*P(C)*P(D)

27. Conditional Probability P(A|B) Formula
P(A|B) = P(A  B) / P(B), if P(B) > 0
(Note that this is an algebraic manipulation of the formula for the probability of the intersection of 2 events.)
i.e. the conditional probability is the probability that both occur divided by what is given occurs