Probability Notes

Math 309

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; )

Since events are sets, we need to understand the basic set operations

• Intersection (AB) - everything in A and B

• Union (A B) - everything in A or B or both

• Complement (AC ) - everything not in A

• Difference A – B = A BC – everything in A that is not in B

• 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

DeMorgan’s Laws

• (A B)C = (AC BC)

• (A B)C = (AC BC)

• Convince yourself that these are reasonable with Venn diagrams!

Another definition -

A and B are mutually exclusive iff

A B =

Axioms of Probability(these are FACT, no proof needed!)

Let A represent an event, S the sample space,

• P(S) = 1• For pairwise mutually exclusive events, the

probability of their union is the sum of their respective probabilities, i.e.

P(A1A2 . . . An . . .) =

P(A1)+P(A2)+ . . . +P(An) + . . .

0 ( ) 1P A

From Axiom 3, it can be shown that:(Prop. 1*)

• Let {A1, A2, . . . ,An} be a mutually exclusive set of events. Then

P(A1A2 . . . An) = P(A1) + P(A2) + . . . + P(An)

• Let A and B be mutually exclusive, our last theorem with n = 2 gives:

P(A B) = P(A) +P(B)

•

More Theorems (Propositions)

Let A and B be any two events.

• Prop. 4.1 - P(AC ) = 1 - P(A)

• Prop. 4.2 - If A is a subset of B, then P(A) <= P(B)

• Prop. 4.3 - P(A B) = P(A) + P(B) - P(A B)

• Prop. 2* - P(A) = P(A B) + P(A BC)

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)

A C

B

Sample Spaces with Equally Likely Outcomes

In an experiment where all sample points are equally likely, one can find the probability of an event by counting two sets.

# of sample points in P(A)

# of sample points in S

A

Combinatorial Methods

Math 309

Chapter 1

Combinatorics

• Basic Principle of Counting– (a.k.a. Multiplication Principle)

• Permutations– Permutations with indistinguishable objects

• Combinations

Basic Counting Principle

• If experiment 1 has m outcomes and experiment 2 has n outcomes, then there are mn 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.

• 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.

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)!

Permutations with Indistinguishable Objects

• Order the objects as if they were distinguishable

• Then “divide out” those arrangements that look identical.

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!

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

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

Conditional Probability and Independence

Math 309

Chapter 3

Conditional Probability P(A|B)

• P(A|B) is read, “the probability of A given B”

• B is known to occur.

• P(A|B) = P(A B) / P(B), if P(B) > 0

• i.e. the conditional probability is the probability that both occur divided by what is given occurs

The multiplication rule and intersectionmultiply

• P(A B) = P(A)*P(B|A) • = P(B)*P(A|B)• (Note that this is an algebraic manipulation of the formula for conditional probability.)

• Intersections get more complicated when there are more events, e.g.

• P(ABCD)• = P(A)* P(B|A)*P(C|AB)*P(D|A BC)

Independent Events

• A and B are independent if any of the following are true:– P(AB) = 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 (AB C D) = P(A)*P(B)*P(C)*P(D)