# Probability

## Rules

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

If A and B are independent then:

P(A ∪ B) = P(A) + P(B)

### Product Rule

P(A ∩ B) = P(A|B) * P(B)

If A and B are independent then:

P(A ∪ B) = P(A) * P(B)

Product rule can also be written as Bayes’ theorem (baby version):

P(A|B) = (P(A ∩ B)) / (P(B))

## Event Relationships

Two events A and B can be:

• Dependent - If A occurs it affects P(B) or vice-versa
• Independent - If A occurs it does not affect P(B) or vice-versa
• Disjoint - A and B are mutually exclusive. Always dependent
• Complementary - A and B are are the only two possible (disjoint) events of the same random process

A and B disjoint =>

 P(A ∩ B) = 0

A and B complementary =>

P(A) + P(B) = 1

## Bayesian Inference

Posterior probability - P(hypothesis | data)

TODO - bayesian Inference

## Binomial Distribution

A random variable has binomial distribution when:

1. Trials are independent
2. The number of trials is fixed
3. Only two possible outcomes (success / failure)
4. P(success) is the same for each trial

### Probability of k successes in n trials

((n),(k)) * p^k * (1 - p)^(n - k)

#### Binomial coefficient

n choose k

((n),(k)) = (n!) / (k! (n-k)!)

### Expected number of successes

#### Mean

μ = n * p

#### Standard deviation

σ = sqrt(n * p * (1-p))

### Normal Distribution Approximation to Binomial

When n is sufficienly large, the binomial distribution can be approximated by the normal distribution.

Rule of thumb for “sufficienly large”:

n * p ≥ 10, n* (1 − p) ≥ 10