# Sample Space **Common letters** $S$ (for sample space) $\Omega$ (for multiverse) **Words Definition** The collection of **all possible things that could happen.** **Symbols Definition** A set of elements. These are denoted with curly brackets to show the things that can happen $S = \{ ... \}$ where we would write the possible things in $S$ inside the curly brackets. **Example (Roll Two 4-Sided Dice)** If we roll two possible dice, then the set of possible outcomes is $S = \left\{ \begin{array}{cccc} (1,1), & (1,2), & (1,3), & (1,4),\\ (2,1), & (2,2), & (2,3), & (2,4),\\ (3,1), & (3,2), & (3,3), & (3,4),\\ (4,1), & (4,2), & (4,3), & (4,4) \end{array}\right\}$ $S$ is a set with 16 elements in it. Each element is an ordered pair $(x,y)$ with both $x,y$ between 1 and 4. This is sometimes written as the Cartesian product $\times$ of set as $S = \{1,2,3,4\}\times\{1,2,3,4\}$. # **Simple Event** (aka **Outcome** aka Atom) **Common letters** $e$ or $o$ $\omega$ (which is the lowercase version of the Greek letter $\Omega$) **Words Definition** A specific way things actually happened. **Symbols Definition** One of the elements of the sample space. $e \in S$ **Example (Roll Two 4-Sided Dice)** $e = (3,1) \in S$ is the outcome of rolling exactly a $3$ on the first dice and exactly a $1$ on the second dice. # **Event** **Common Letters** $A$ or $E$. Since there are so many different events in probability, and you need one letter per event, we end up using many different letters for these **Words Definition** Something that either happens or doesn't happen. The event $A$ is the collection of outcomes where "$A$ happens". **Symbols Definition** Any[^1] subset of the sample space, $A \subset S$ This corresponds to the set of outcomes where "$A$ happens." **Example (Roll Two 4-Sided Dice)** Let $A$ be the event that the sum of the dice is exactly 4. Then one can find that $A = \{ (1,3), (2,2), (3,1) \} \subset S$ Let $B$ be the event that the two dice are equal. Then one can find that $B = \{ (1,1),(2,2),(3,3),(4,4) \} \subset S$ # Intersection aka **"and"** **Common notation** $\cap$ **Words Definition** A new event made from the events $A$, $B$. Both condition $A$ *and* condition $B$ happens. **Symbols Definition** $A \cap B$ , the intersection of events $A$ with $B$. $A \cap B = \{e : e\in A\text{ and }e\in B\}$ ## Picture of Intersection $\cap$ ![[Venn0001.svg]] **Example (Roll Two 4-Sided Dice)** Let $A$ be the event that the sum of the dice is exactly 4 and let $B$ the event that the two dice are equal. Then one can find that $A \cap B = \{(2,2)\}$. This is also equal to the event that the dice come exactly as 2 and 2. # Union aka "or" **Common Notation** $\cup$ **Words Definition** A new event made from the events $A$, $B$. Either condition $A$ *or* condition $B$ happens (or both[^2]). **Symbols Definition** $A\cup B$ , the union of events $A$ with $B$. $A \cup B = \{e : e\in A\text{ or }e\in B\text{ or both}\}$ ## Picture of Union $\cup$ ![[Venn0111.svg]] **Example (Roll Two 4-Sided Dice)** Let $A$ be the event that the sum of the dice is exactly 4 and let $B$ the event that the two dice are equal. $A \cup B = \{(1,3),(3,1),(1,1),(2,2),(3,3),(4,4)\}$ are the ways the sum can be either 4 or the two dice are equal. # Mutually Exclusive aka Disjoint **Words Definition** The events $A$ and $B$ have *nothing in common.* It is impossible for both $A$ and $B$ to happen at the same time **Symbols Definition** $A \cap B = \emptyset$ i.e. the two sets have nothing in their intersection. The symbol $\emptyset$ means "empty set" which stands for the set with nothing in it. **Is it a complement?** The complement is a special type of disjoint set. See complements for more. **Example (Roll Two 4-Sided Dice)** Let $A$ be the event the first dice is $4$ and let $B$ be the event that the sum of the dice is 3. Then $A$ and $B$ are disjoint. # **Independent** To be defined later but do **not** mix it up with disjoint! Disjoint and independent sound similar but mean completely different things in probability class. # Complement aka "not" **Common Notation** For an event $A$, the complement is written as $A^c$ (or sometimes $A^\prime$ or $\bar{A}$ or simply $\text{not }A$). **Words Definition** For an event $A$, its the opposite of $A$ happening. Its the event "$A$ does not happen" **Symbols Definition** It is the set of things which are not in $A$: $A^c = \{ e : e \not \in A\}$ **How to recognize a complement?** Note that this is disjoint from $A$ and also $A$ and $A^c$ together are the entire space $S$. i.e. symboliccaly, we write that $A^c$ satisfies two things: $A^c \cap A = \emptyset$$A^c \cup A = S$ These two are the the defining characteristics of the complement, so are a good way to check if something is a complement. Are they disjoint? Together, do they catch every possible outcome? **Example (Roll Two 4-Sided Dice)** ** Let $A$ be the event that the first dice is $4$. Then $A^c$ is the event that the first dice is $1$, $2$, or $3$. # Why do we care? Probability rules for combining events In many probability problems, to get to what you want it is easier to write what you want in terms of unions/intersections/complements and then find the probabilities of the individual things. Then we can use rules like: $\mathbb{P}(A^c) = 1 - \mathbb{P}(A)$ For example, in the Birthday Problem, it is easier to find the probability that two people *don't* share a birthday, than the probability they *do* share a birthday. Another rule or disjoint events: $ \mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B) \text{ when }A,B\text{ are disjoint} $ and more generally, the inclusion-exclusion rule says: $\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B) - \mathbb{P}(A \cap B)$ ## Lecture Recording and Blackboard See also the PDF "Blackboard" [[Attachments/Basic_Probability_Definitions.pdf|Basic_Probability_Definitions]] from this Lecture with these basic definitions: https://youtu.be/s9PNeEYumG0 [^1]: For infinite probability spaces, it turns out that not every possible subset is allowed as an event. In those cases, there is a special set of "allowed events". Don't worry about this if you are dealing with finitely many outcomes! [^2]: Note that in math, the "or both" is implied and often not written. Do not mix this up with the English language use of "or" which often means "exactly one of". That is written as "xor" in math, standing for "exclusive or"