1. Probability Basics

Went over basic Set Theory too - 1. Set Theory

Sample space $S$ - the set of all possible outcomes of an experiment

An event $A$ is a subset of the sample space $S$

disjoint - unique or not intersecting or something

Theorems

probability of the sample space must sum to 1

probability of any event must be between 0 and 1

the probability of the union of all the events is equal to the summation of the probabilities of each event

If $A\cap B=\varnothing$, then $P(A\cup B)=P(A)+P(B)$

For any 2 events, $P(A\cup B)=P(A)+P(B)-P(A\cap B)$

Why does Inclusion Exclusion add the intersection back?

Because it counts the subtracted intersect probability’s center in the venn diagram by 1 more than it should.

Theorem 5: If $A\subseteq B$, then $P(A)\le P(B)$

If all outcomes are equally likely, a simple sample space $SSS$, then for any event in a $SSS$, P(A) = \frac{\text{# elements in A}}{\text{# elements in S}}

Counting

Multiplication rule

number of outcomes for flipping a coin 3 times

• $2\cdot 2\cdot 2=6$

Choose 2 red socks out of 10 socks, where only 2 socks are red.

• $\frac{2\cdot 1}{10\cdot 9}=\frac{1}{45}$

Permutations

A permutation is the arrangement of $n$ items in a specific order. (order matters)

$n$ choose $r$.

Combinations

Order does not matter; therefore, the # number is smaller.

Add r! to denominator

$P=C$ if $r=1$

Theorems

$\text{N choose r = N choose N - R}$

Sampling

Without replacement

$a$ objects of type A and $b$ objects of type B

Given types A and B and that choose K of type A, choose N total

A choose K times B choose n-k 
divided by
# of ways to choose n out of a + b

Independence vs Disjoint

Independence = one event does not affect the chances of the other happening Disjoint = Both events cannot share an outcome

$P(A\mid B)=P(A)$

$P(A\cap B)=P(A)P(B)$

Series

Geometric = $a+r$