### Revision Lectures ##### R.1 Definitions & Notation - **Random experiment** – any experiment whose exact outcome cannot be predicted with certainty - **Outcome** – the result of a random experiment - **Event** – a set of possible outcomes - **Sample space** – the set containing all possible outcomes - **Probability** of an event $E$, $\mathbb{P}(E)$ – a number measuring how likely event $E$ is from a random experiment. Usually, this is interpreted as the long run frequency of $E$ in repeat experiments

The probability of an event depends on the nature of the experiment.

**Set Notation** - **Union** - $A\cup B=$ event "either A or B occurs" - **Intersection** - $A\cap B=$ event "both A and B occur" - **Complement** - $A^c=$ event "A does not occur" (=$A'$) **Subset Notation** - $A\subseteq B\Rightarrow$ A implies B - $A\cap B=\phi\Rightarrow$ mutually exclusive events (they cannot occur together) **De Morgan's Laws** - $(A\cup B)^c=A^c\cap B^c$ - $(A\cap B)^c=A^c\cup B^c$

Depict probability using Venn Diagrams.

*** ##### R.2 Probability Rules Any probability measure should satisfy Kolmogorov's Axioms. > [!NOTE] Kolmogorov's Probability Axioms > The probability measure $\mathbb{P}(\cdot)$ satisfies: > 1. $0\leq\mathbb{P}(E)\leq 1$ for any event $E$ > ==Probability must be between 0 and 1.== >
> 2. $\mathbb{P}(S)=1$ > ==The chance that some event would happen from the set of all possible outcomes is 1. It is guaranteed one of them would happen.== >
> 3. For any (infinite) sequence of mutually exclusive events $E_1,\ E_2,\ \dots,$ > $$ > \mathbb{P} \left( \bigcup\limits_{i=1}^\infty E_i \right) = \sum\limits_{i=1}^\infty \mathbb{P}\left(E_i\right) > $$ From these axioms, - If $A$ and $B$ are mutually exclusive, $\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)$. - $\mathbb{P}(A^c)=1-\mathbb{P}(A)$ - The **additive law**, $\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cap B)$ *** ##### R.3 Equally Likely Outcomes If all outcomes are equally likely, $$\mathbb{P}(E)=\frac{\text{number of outcomes in }E}{\text{total number of possible outcomes}}$$ **Rules** 1. **Multiplication rule** – If there are $k$ steps in obtaining an outcome, and $n_i$ ways of completing step $i$, then the total number of outcomes is $$n_1\times n_2\times\dots\times n_k$$ 2. **Permutation** – An ordered sequence of the elements in a set. The number of different permutations of $n$ elements is $$P_n=n\times(n-1)\times(n-2)\times\dots\times2\times1=n!$$ 3. **Combination** - A subset of elements selected from a larger set. The number of combinations of size $r$ that can be selected from a set of $n$ elements is $$\pmatrix{n\\ r}=C^n_r=\frac{n!}{r!(n-r)!}$$ *** ##### R.4 Conditional Probability The **conditional probability** of $B$, conditional on $A$ if $\mathbb{P}(A)>0$, is defined as $$ \mathbb{P}(B \mid A) = \dfrac{\mathbb{P}(A \cap B)}{\mathbb{P}(A)}=\text{probability of }B\text{ given that }A\text{ has occurred.}$$ Ordering is important – $\mathbb{P}(\mathbb{B\mid A})\text{ vs }\mathbb{P}(\mathbb{A\mid B})$

Depict conditional probability using Tree Diagrams.

![[Screenshot 2025-06-04 at 12.29.27 pm.png|center|200]] *** ##### R.5 Conditional Probability Rules - If $B \subseteq A,\ \mathbb{P}(A \mid B) = 1$ - **Multiplication law** – $\mathbb{P}(A \cap B) = \mathbb{P}(A \mid B) \times \mathbb{P}(B)$ - **Law of total probability** – $\mathbb{P}(A) = \mathbb{P}(A \mid B) \times \mathbb{P}(B) + \mathbb{P}(A \mid B^c) \times \mathbb{P}(B^c)$ - **Bayes' rule** – $\mathbb{P}(A) > 0$ and $\mathbb{P}(B) > 0$, then $\mathbb{P}(A \mid B) = \mathbb{P}(B \mid A) \times \dfrac{\mathbb{P}(A)}{\mathbb{P}(B)}$

Bayes' rule reverses the order of conditioning.

*** ##### R.6 Independence of Two Events If two events $A$ and $B$ are unrelated, knowing whether or not $B$ happens doesn't change the chance of $A$ happening. Independence implies that $$\mathbb{P}(A\mid B)=\mathbb{P}(A)$$ Two events $A$ and $B$ are said to be **independent** if and only if $$\mathbb{P}(A\cap B)=\mathbb{P}(A)\times\mathbb{P}(B)$$ > [!NOTE] > You can only multiply the probability of a set of events to get the joint probability if the events are independent. As in, $\mathbb{P}(A\cap B)=\mathbb{P}(A)\times\mathbb{P}(B)$. **Applications of Independence** - Study if two variables are related to each other – to **test for independence** - **Random sampling** – ensures that observations are independent of each other - **Parallel back-ups** – ensuring a system has an independent backup reduces chance of system failure - **False positives** – If you independently check for a problem many times, with a non-zero chance of a false positive each time, false positives accumulate quickly

MATH2859