---
description: The Language of Probability (BH Chapter 1)
---

# Section 1

Like the English language, the language of probability has its own nouns, verbs, and adjectives. Confusing these parts of speech will result in "category errors." Let's familiarize ourselves with these parts of speech.

* **Probability** will always be a _value between 0 and 1 inclusive_ that expresses how likely a particular event will occur out of all possible outcomes in the sample space.  Moreover, $$P(\Omega) = 1$$.
* **Experiment**: An experiment is a process of obtaining outcomes about an uncertain phenomenon.
* **Sample space** ($$\Omega$$): A sample space contains all the possible experimental outcomes that could happen.
* **Event**: An event is a certain subset of the sample space.
* **Naive Definition** - If all outcomes are equally likely, the probability of event {A} happening is

$$
P_{{naive}}(A) = \frac{{number\ of\ outcomes\ favorable\ to\ {A}}}{{number\ of\ outcomes}}
$$

## Set Theory

**Intersection** - Given two events A and B, A ∩ B means A _and_ B.

**Union** -Given two events A and B, A ∪ B means A _or_ B.

**Complement** - Given an event A, $$A^C$$ is called A’s complement, and means when "A does _not_ occur", everything that’s not in A.

**Subevent** - Given two events A and B, _A_ ⊆ _B_ means "B includes everything in A". We can write all valid events A as a subevent of the total sample space Ω: _A_ ⊆ Ω.

**De Morgan’s Laws** - A useful identity that can make calculations easier by relating unions to intersections. Analogous results hold with more than two sets.

$$
\begin{array}{l}{(A \cup B)^{c}=A^{c} \cap B^{c}} \end{array}
$$

$$
\begin{array}{l}{ (A \cap B)^{c}=A^{c} \cup B^{c}}\end{array}
$$

**Principle of Inclusion-Exclusion** - For any events $$A_1 ,\dots , A_n$$,

$$
P\left(\bigcup_{i=1}^{n} A_{i}\right)=\sum_{i} P\left(A_{i}\right)-\sum_{i<j} P\left(A_{i} \cap A_{j}\right)+\cdots+(-1)^{n+1} P\left(A_{1} \cap \cdots \cap A_{n}\right)
$$

## Counting

**Multiplication Rule** - If we have _n_ decisions to make and the _j_-th decision has $$r_j$$outcomes, then the total number of potential outcomes is $$r_1\cdot r_2\cdot\dots\cdot r_{n-1}\cdot r_n$$

**Factorial** - The number of ways to order _n_ objects is given as $$n! = n\cdot(n-1)\cdot\dots\cdot2\cdot1$$

**Binomial Coefficient Formula** - For _k_ ≤ _n_, we have

#### $$\binom{n}{k} = \frac{n(n-1)\dots(n-k+1)}{k!} = \frac{n!}{k!(n-k)!}$$

$$\binom{n}{k} = \binom{n}{n-k}$$

**Binomial Theorem** - $$(x+y)^n=\sum_{k=0}^n \binom{n}{k}x^k y^{n-k}$$

The sampling table gives the number of possible samples of size $$k$$ out of a population of size $$n$$, under various assumptions about how the sample is collected.

$$
\begin{array}{c|cc} 
& \text { Order Matters } & \text { Order Doesn't Matter } \\
\hline \text { With Replacement } & n^{k} & \left(\begin{array}{c}
n+k-1 \\
k
\end{array}\right) \\
\text { Without Replacement } & \frac{n !}{(n-k) !} & \left(\begin{array}{l}
n \\
k
\end{array}\right)
\end{array}
$$
