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Week-2

Math 183 • Statistical Methods • Spring 2026

Siddharth Vishwanath

Learning objectives

Randomness
Sample spaces and Events
Elements of set theory
Probability
Random variables

Example

Example

You’ve decided to drive to campus today, and you’re just about on time for your lecture at 8:00am. You’re on the home stretch, and made it all the way to the Osler Parking Structure. What happens next?

Example

Example

The there are free spots available 🎉

The parking lot is jammed up the wazoo 🤕

The parking lot is closed for construction 🤯
…

These are examples of events
The event you actually observe on the given day is (sort of) random

Some Definitions

Random phenomenon

A random phenomenon is a situation in which we know what outcomes could possibly occur, but we don’t know for sure which particular outcome will happen

Trials & Outcomes

Each occasion upon which we observe a random phenomenon is called a trial. A particular observation which we record from a trial is called the trial’s outcome.

Events & Sample Space

An event is a combination of outcomes we might observe from a random phenomenon. We call the collection of all possible events the sample space.

Example

Coin toss

You toss a coin with two sides H and T. What is the random phenomenon? What is the trial here? What is the sample space?

When you toss the coin, it lands with T facing up. What is the outcome? What event have you just observed?

Example

Deck of Cards

You shuffle a full deck of cards, and draw a card from the deck. What is the random phenomenon? What is the trial here? What is the sample space?

The card you draw is the Queen of hearts. What is the outcome of your trial? What event have you just observed?

Probability

Probability

The probability of an event quantifies the likelihood of observing the event as the outcome of a trial among the universe of all possible events in the sample space.

Types of Probability

Empirical Probability:

The probability of an event based on repeatedly observing the outcome of an event

Subjective Probability:

The probability guesstimate you have about the possibility of an outcome for an event

Theoretical Probability:

The probability of an event based on an actual mathematical model \[\mathbb{P}(A) = \frac{\# \text{outcomes in }A}{\# \text{all possible outcomes}}\]

Three rules for working with probability

Make a picture
Make a picture
Make a picture

Bonus rule

Make a picture

The event $A \subset S$

images courtesy: Probability Tree, Siegel & Wagner

\[ {\mathbb P}(A) = \frac{\# \text{outcomes in }A}{\# \text{all possible outcomes}} = \frac{\text{size of } A}{\text{size of } S} \]

Revision of Set Theory

Given an event $A$, the event “not A” is given by

\[ A^c = S \setminus A \]

Example

Let $A = \left\{♠, ♣\right\}$ be the event that a card randomly drawn is a club or a spade. What is the complement of this event?

Revision of Set Theory

Given two events $A$ and $B$

the event “A AND B” is given by \[A \cap B\]

Example

Flip a coin four times. Let $A$ be the event that the first two tosses are H. Let $B$ be the event that the every even coin toss is a H. What is the event $A$ AND $B$?

Revision of Set Theory

Given two events $A$ and $B$

the event “A OR B” is given by \[A \cup B\]

Example

Flip a coin four times. Let $A$ be the event that the first two tosses are H. Let $B$ be the event that the every even coin toss is a H. What is the event $A$ OR $B$?

Independence

Consider the following two events

\[ A = \left\{\text{Mr. Beast uploads a new video on YouTube}\right\}\\ B = \left\{\text{Selena Gomez DMs you on Instagram}\right\} \] If $B$ actually happens, how does it affect the probability of $A$?

Independent Events

Two events $A, B$ are said to be independent if and only if \[ {\mathbb P}(A \cap B) = {\mathbb P}(A) \times {\mathbb P}(B) \]

Informally, two events are independent if and only if the outcome of one event doesn’t affect the probability of the other.

Axioms of Probability

The probability of an event is between $0$ and $1$, i.e., for any event $A \subset S$ \[ 0 \le {\mathbb P}(A) \le 1 \]

The set of all possible outcomes of a trial must have probability $1$, i.e., \[ {\mathbb P}(S) = 1 \]

The probability of an event $A$ not occurring is \[ {\mathbb P}(A^c) = 1 - {\mathbb P}(A) \]

For two disjoint events $A$ and $B$ (i.e., $A \cap B = \varnothing$), the probability of $A$ or $B$ is their sum, i.e., \[ {\mathbb P}(A \cup B) = {\mathbb P}(A) + {\mathbb P}(B) \hspace{2em} \Leftrightarrow \hspace{2em} A \cap B = \varnothing \]

When the events $A$ and $B$ are not disjoint (i.e., $A \cap B \neq \varnothing$), the probability of $A$ or $B$ is \[ {\mathbb P}(A \cup B) = {\mathbb P}(A) + {\mathbb P}(B) - {\mathbb P}(A \cap B) \]

When the events $A$ and $B$ are independent, the probability of $A$ and $B$ is \[ {\mathbb P}(A \cap B) = {\mathbb P}(A) \times {\mathbb P}(B) \]

Rule 5: Inclusion-Exclusion Principle

Inclusion Exclusion Principle

\[ \begin{aligned} {\mathbb P}(A \cup B \cup C) &= {\mathbb P}(A) + {\mathbb P}(B) + {\mathbb P}(C) \\ &- {\mathbb P}(A \cap B) - {\mathbb P}(A \cap C) - {\mathbb P}(B \cap C) \\ &+ {\mathbb P}(A \cap B \cap C) \end{aligned} \]

Be Careful

Beware of probabilities that don’t add up to 1.
- The sum of the probabilities for all possible outcomes must total 1
- If the sum is less than $1$, make sure have accounted for all possible events
- If the sum is greater than $1$, you may be assuming some events are disjoint when they are not
Don’t add probabilities of events if they’re not disjoint.
- Remember the inclusion-exclusion principle
Don’t multiply probabilities of events if they’re not independent.
Don’t confuse disjoint and independent.

Examples

Getting to School

Your first class on M/W/F is a 8:00AM lecture at Pepper Canyon Hall. You like to cut things fine, and consistently leave home to catch the 7:40AM blue line trolley from Nobel Dr to get to class. There’s a 20% chance the trolley is late on any given day.

What is the probability that you’re late to class on any given day?
What is the probability that the first time you’re late this week is a Friday?
What is the probability that you’re late every day this coming week?
What is the probability that you’re late at least once this coming week?

The Enchanted Portal

While playing Dungeons & Dragons, your party encounters an enchanted portal. To unlock this portal and proceed to the treasure room, a specific combination of numbers from three dice must be rolled. These aren’t ordinary dice; they are a D4 (a four-sided die), a D6 (a six-sided die), and a D8 (an eight-sided die).

The Enchantment’s Conditions: (i) D4 must show a prime number, (ii) D6 should not show an even number, and (iii) D8 must show a number greater than 5.

Formulate the events based on the conditions given.
Calculate the probability of each event occurring.
Are the events independent?
What is the probability that all three conditions are satisfied simultaneously?
What is the probability that at least one of the conditions is satisfied?

Cell phone survey

A 2010 study conducted by the National Center for Health Statistics found that 25% of U.S. households had no landline service. This raises concerns about the accuracy of certain surveys, as they depend on random-digit dialing to households via landlines.

We are going to pick five U.S. households at random:

What is the probability that all five of them have a landline?
What is the probability that at least one of them does not have a landline?
What is the probability that at least one of them does have a landline?

Social media survey

	Instragram	No Instagram
TikTok	150	50
No TikTok	700	…

A survey on $1000$ UCSD undergrads resulted in the above table.

For a randomly selected undergrad:

What is the probability that they use Instagram?
What is the probability that they use TikTok?
What is the probability that they use both Instagram and TikTok?
What is the probability that they use either Instagram or TikTok?
What is the probability that they use Instagram or TikTok, but not both?
What is the probability that they don’t use Instagram or TikTok?
Are the events that they use Instagram and TikTok independent?

Traffic

${\mathbb P}(\text{traffic jam}) = 0.20$
${\mathbb P}(\text{traffic jam}) = 0.50$
${\mathbb P}(\text{traffic jam}) = 0.75$
${\mathbb P}(\text{traffic jam}) = 0.10$

What is the probability of encountering a traffic jam from start to finish?

Law of Total Probability

Law of Total Probability

Let $A$ and $B$ be two events. Then \[ {\mathbb P}(A) = {\mathbb P}(A \cap B) + {\mathbb P}(A \cap B^c) \]

Traffic

${\mathbb P}(\text{traffic jam}) = 0.20$
${\mathbb P}(\text{traffic jam}) = 0.50$
${\mathbb P}(\text{traffic jam}) = 0.75$
${\mathbb P}(\text{traffic jam}) = 0.10$

\[ \begin{aligned} {\mathbb P}(\text{traffic jam}) &= {\mathbb P}(\text{traffic jam} \cap \text{R}_1) + {\mathbb P}(\text{traffic jam} \cap \text{R}_1^c) \end{aligned} \]

\[ {\mathbb P}(\text{traffic jam}) = {\mathbb P}\left(\{\color{red}{\text{red}} \cup \color{gold}{\text{yellow}}\} \cap \{\text{R}_1\}\right) + {\mathbb P}\left(\{\color{blue}{\text{blue}} \cup \color{purple}{\text{purple}}\} \cap \{\text{R}_1\}^c\right) \]

Conditional Probability

Conditioning on the event $A$ means that we operate under the assumption that $A$ happens, meaning, the sample space reduces to everything under $A$.

$B | A$ is the event that $B$ happens when we know that $A$ has already occurred.

\[{\mathbb P}(B| A) = \frac{{\mathbb P}(A \cap B)}{{\mathbb P}(A)}\]

Rearranging, we also get the general expression \[{\mathbb P}(A \cap B) = {\mathbb P}(B | A) \times {\mathbb P}(A)\]

Alternate definition for Independence

$A$ and $B$ are independent if and only if ${\mathbb P}(B | A) = {\mathbb P}(B)$

Blindfolded Rolling Dice

You roll two dice blindfolded. I have a look at the outcome and tell you that their sum is a prime number.

What is the probability that both dice landed on the same number?
What is the probability that both dice landed on an even number?
What is the probability that both dice landed on an odd number?
What is the probability that one of them is a “3”?

Traffic

${\mathbb P}(\text{traffic jam}) = 0.20$
${\mathbb P}(\text{traffic jam}) = 0.50$
${\mathbb P}(\text{traffic jam}) = 0.75$
${\mathbb P}(\text{traffic jam}) = 0.10$

\[ \begin{aligned} {\mathbb P}\left(\{\color{red}{\text{red}} \cup \color{gold}{\text{yellow}}\} \cap \text{R}_1\right) &= {\mathbb P}\left(\{\color{red}{\text{red}} \cup \color{gold}{\text{yellow}}\} \;\vert\; \text{R}_1\right) \times {\mathbb P}(\text{R}_1)\\ &= (\color{red}{0.2} + \color{gold}{0.5} - \color{red}{0.2} \times \color{gold}{0.5}) \times 0.5\\ \\ {\mathbb P}\left(\{\color{blue}{\text{blue}} \cup \color{purple}{\text{purple}}\} \cap \text{R}_1^c\right) &= {\mathbb P}\left(\{\color{blue}{\text{blue}} \cup \color{purple}{\text{purple}}\} \;\vert\; \text{R}_1^c\right) \times {\mathbb P}(\text{R}_1^c)\\ &= ( \color{blue}{0.75} + \color{purple}{0.1} - \color{blue}{0.75} \times \color{purple}{0.1}) \times 0.5 \end{aligned} \]

Conditional Probability Rules

Let $A$ and $B$ be two events

Law of Total Probability

\[ {\mathbb P}(B) = {\mathbb P}(B | A) \times {\mathbb P}(A) + {\mathbb P}(B | A^c) \times {\mathbb P}(A^c) \]

Bayes’ Theorem

\[ {\mathbb P}(A | B) = \frac{{\mathbb P}(B | A) \times {\mathbb P}(A)}{{\mathbb P}(B)} \]

Visualizing Conditional Probabilities

Example

	💉✅	💉❌
COVID 🤒	500	1200
No COVID 🙂	4500	4800

What is the probability of getting COVID conditioned on the event that you got the vaccine?

$A =$ {🤒}, $B =$ {💉✅}. $\ \ \ \ \ \ A | B = A \textbf{ given } B =$ {🤒 after 💉✅}.

\[ \begin{aligned} {\mathbb P}(A | B) &= \frac{\#\left\{🤒 \text{ AND } 💉✅\right\}}{\#\left\{🤒 \text{ AND } 💉✅\right\} + \#\left\{🙂 \text{ AND } 💉✅\right\}}\\ \\ &= \frac{500}{500 + 4500} \end{aligned} \]

Independence vs. Disjoint

Food for thought

We know that $A$ and $B$ are two independent events. Does it imply that $A$ and $B$ are disjoint, i.e., $A \cap B = \varnothing$?

Examples

Example

Marbles

A box contains $8$ white marbles and $3$ red marbles. We pick draw two marbles at random from the box (without replacing them). What is the probability that the second marble selected is red, given that the first marble selected is white?

Netflicks streaming

A study by Netflicks revealed that at 80% of the students at UCSD stream Netflicks to watch movies and TV. The study further revealed that only 20% of the students who watch Netflicks actually pay for it themselves.

We are going to pick a student at random.

Formulate the two events mentioned in this scenario.
In mathematical terms, define the event where the student watches Netflicks but doesn’t pay for it.
What is the probability of this event?

The mystery of the dead plant

You have a plant (🪴) which needs watering on a regular basis. Without water the plant will die with probability $0.7$, and with water it will die with probability $0.1$.

You travel out of town and and ask your (absent minded) friend to water the plant. Your friend, being absent minded, sometimes forgets instructions they are given with probability $0.2$.

a. What is the probability that the plant will die?

b. You come back from your trip and find that the plant is dead. What is the probability that your friend forgot to water the plant?

Highway Safety

A recent Maryland highway safety study found that in 77% of all accidents the driver was wearing a seatbelt. Accident reports indicated that 92% of those drivers escaped serious injury (defined as hospitalization or death), but only 63% of the nonbelted drivers were so fortunate.

What’s the probability that a driver who was seriously injured wasn’t wearing a seatbelt?

COVID-19

A recent study found that 80% of the people who tested positive for COVID-19 had a fever. The study also found that 95% of the people who tested negative for COVID-19 did not have a fever.

a. What is the probability that a person has COVID-19 given that they have a fever?

b. What is the probability that a person has COVID-19 given that they do not have a fever?