“Machine Learning, Deep Learning, Data Science, etc. are all in the same nexus – which is the basically Probability plus Statistics. ”

Let’s fact it: life is uncertain. But one thing is certain: We need a way to make predictions and make decisions under uncertainty.

Conditional Probability

Bonferroni's inequality

Discrete random variables

Continuous random variables

Further topics

Bayesian inference

Limit theorems and statistics

Random arrival processes

Markov chains

Describe possible outcomes

Describe beliefs about likelihood of outcomes

**Sets:**A collection of distinct element$$\begin{array}{c}\text{finite}:\{a,b,c,d\}\\ \text{infinite}:x\in \mathbb{Re}\end{array}$$

Namely, the elements should be **mutually exclusive** and **collectively exhaustive**.

**Mutually exclusive** means that, two or more events that cannot happen simultaneously

Being **collectively exhaustive** means something else-- that, together, all of these elements of the set exhaust all the possibilities, which is

Event

**Complement of A** :

Nonnegativity:

$P(A)>0$ Normalization:

$P(\mathrm{\Omega})=1$ Additivity of disjoint events (finite, countable set):

$If\text{}A\cap B=\varnothing ,\text{}then\text{}P(A\cap B)=P(A)+P(B)$

Probability, at the minimum, gives us some rules for thinking systematically about uncertain situations.

$B=A\cup (B\cap {A}^{c})$ Non-disjoint Events:

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$ More consequences:

$P(A\cup B\cup C)=P(A)+P({A}^{c}\cap B)+P({A}^{c}\cap {B}^{c}\cap C)$

Union bound:

$P(A\cup B)\le P(A)+P(B)$

The probability of union **not** equal to the sum of the

Loosely speaking, probabilities can be interpreted as “**Frequency**” or “**Describing our beliefs**”.

And if we had a partition of our sample space into an infinite sequence of event **total probability theorem** or “weighted average”:

**Example of total probability theorem**

Assume we have

Set event

If we take the intersection of two sets and then take the complement of this intersection, what we obtain is the union of the complements of the two sets.

Or generally we have:

**Problem Solving Examples**

Set in A not in B :

Giving

So the whole term equal to “**union - intersection**”.

Utilize to interpret the union bound.

And Vise versa, now they are most:

(b) Generalize to the case of

Proof:

We initial beliefs

**Model**of the world under each${A}_{i}:$ $\mathrm{P}(B\mid {A}_{i})$ Under each particular situation, the model tells us how likely event

is to occur.$B$ **Inference**to draw conclusions about causes:$\mathrm{P}({A}_{i}\mid B)$ If we actually observe that B occurred, then we use that information to draw conclusions about the possible causes of

$B$

Romeo and Juliet arrive with a delay between 0 and 1 hour

with all pairs of delays being “equally likely," that is, according to a uniform probability law on the unit square.

The first to arrive will wait for 15 minutes and will leave if the other has not arrived.

What is the probability that they will meet?

Draw with Uniform probabilities on a square.

Geometric Demonstration just boils down to not a probability problem, but a problem in geometry - Calculate the area of the space.

Then you can ask, if he wants to have at least a 90% chance of meeting her, how long should he be willing to wait? ?