slan's blog


Study of Probability

Lesson 1: Probability models and axioms

topic details
What is probability? a framework that dealing with uncertainty

How to setup a probability model?
basic rules of
Probability models

Sample space

Discrete Sample space
Sample space, sets
1. list should be mutually exclusive
2. should be collectively exhaustive

You can also use T+ H , raining, + H, not raining in coin flip, it's allowed

Discrete Sample space, Dice experiment
Use a sequential description or tree-based description

it's 2 stage.
It's has a sample space which is finite.
Continuous Sample space sample space is infinite.

assign probabilities to individual outcome has zero probability, so we assign probabilities to subsets
Ground rules 1. probabilities be number between 0 to 1
2. Probabilities should be Non-negativity: $$P(A)>=0$$
3. total probabilities sum=1, $$P(\Omega)=1$$
4. Additivity: if A and B have no common, then $$if A\cap B=\Theta , P(A \cup B) = P(A)+P(B)$$
union of 3 sets,$$ P(A\cup B\cup C) = P((A\cup B)\cup C)=P(A)+P(B)+P(C)$$
it can repeat to any N sets. if A1....Am, disjoint.
subtleties 1. Additivity axiom doesn't quite do the job for everything we would like to do
2. has to do with weird sets, An event is a subset of the sample space, does this mean that we are going to assign probability to every possible subset of the sample space?
-- we'd like to, but not always possible, but we are not going to encounter these sets

laws Discrete uniform law:
if all outcomes are equally likely(N of them ). then p(A)=1/N

Continuous uniform law:
Probability = Area
P(X,Y)=0 , any point area =0

Lesson 2: Conditioning and Bayes' rule

All informations is always partial, what do we do to probabilities if we have some partial informations?

this lesson introduce 3 very useful ways, these ways break problem to simple pieces. infer things we have not seen.

Topic details
Review set up a model, first is come up with list of all possible outcomes. this is a sample space.
1. distinguishable from each other
2. mutually exclusive

how to choose your sample space, depending on how much details you want to capture. its a art.

assign subsets, disjoint subsets behave like masses,

0 probability things is not impossible, its only very very low possible.

problem solving:
- Specify sample space
- Define probability law
- Identify event of interest
- Calculate
New information you set up a model, and somebody give you a New Information, we should revise our beliefs.

$$P(A\mid B) = Probability\ of\ A, given\ that\ B\ occurred$$
How do we revise the probability that A occurs?
Intuitively reasonable way: P(A|B)= part of the A in B.= 2(1+2)

definition: $$P(A\mid B)=\frac{P(A\cap B)}{ P(B)}$$
so P A intersection B is the probability of B times the conditional probability
$$P(A\cap B)= P(B)P(A\mid B) =P(A)P(B\mid A)$$

Lesson 3