Chibany is hungry

Chibany wakes up from dreaming of the delicious meals they will get later today. Twice per day, a student brings them a bento box with a meal as an offering to Chibany. One student brings them a bento box in the early afternoon for lunch and a different student brings them a bento box in the evening for dinner. The meal is either a Hamburger or a Tonkatsu (pork cutlet) . To keep track of their meal possibilities, they list out the four possibilities:

block-beta
    block
        columns 2
        a["H(amburger) H(amburger)"] b["H(amburger) T(onkatsu)"]
        c["T(onkatsu) H(amburger)"] d["T(onkatsu) T(onkatsu)"]
    end

Sets

This forms a set of four elements. A set is a collection of elements or members. In this case, an element is defined by the two meals given to Chibany that day. Sets are defined by the elements they do or do not contain. The elements are listed with commas between them and “$\{$” denotes the start of a set and “$\}$” the end of a set.

Outcome Space

In the context of probability theory, the basic elements of what can occur are called outcomes. Outcomes are the fundamental building blocks that probabilities are built from. As they are fundamental, the Greek letter $\Omega$ is frequently used to refer to this set of possible outcomes. Diligently noting their daily offerings, Chibany defines $\Omega = \{HH, HT, TH, TT \}$. The first letter defines their lunch offering, and the second letter defines their dinner offering. They note that $H$ now always refers to hamburgers and $T$ to tonkatsu.

Why Sets?

Sets are perfect for probability because they let us visualize and count possibilities. Every probability question becomes:

What’s possible? (Define the outcome space)
What am I interested in? (Define the event)
Count both! (Calculate the ratio)

The outcome space and an event will be defined as sets. Probability comes down to the relative sizes of two sets!

This makes probability concrete instead of abstract.

A Note on Unique Elements

Technically, the elements of a set are unique. So, if Chibany writes down getting a pair of hamburgers twice and a hamburger and a tonkatsu ($\{HH, HH, HT\}$), they’ve gotten the same set of possibilities as if they only got one pair of hamburgers and a hamburger and tonkatsu ($\{HH, HT\}$). In other words, $\{HH, HH, HT\} = \{HH, HT\}$.

Think of it like a list where duplicates automatically disappear; only what’s different matters.

For Chibany’s meals

Each element in $\Omega = \{HH, HT, TH, TT\}$ is unique because position matters (first meal vs. second meal). $HT$ ≠ $TH$ because getting tonkatsu for lunch is different from getting it for dinner!

Chibany is skeptical, but will try to keep it in mind. It can be confusing!

Possibilities vs. Events

So far, we have discussed sets, possible outcomes and the set of all possible outcomes $\Omega$. Chibany is interested in the set of possible meals that include Tonkatsu. What is this set?

${HT, TH, TT}$

This is an example of an event. Technically, an event is a set that contains none, some, or all of the possible outcomes.

Events are Subsets

Any event $A$ is a subset of the outcome space $\Omega$. Formally, this is written as $A \subseteq \Omega$.

This means:

Every element in $A$ is also in $\Omega$
$A$ could be empty ($\{\}$, meaning nothing happens)
$A$ could be all of $\Omega$ (something definitely happens)
$A$ could be anything in between

For Chibany’s “contains tonkatsu” event: $A = \{HT, TH, TT\} \subseteq \Omega$

Quick Check

Is $\Omega$ an event?

solution

Yes, it is the event that contains all possible outcomes. This is sometimes called the certain event because something from $\Omega$ must happen.

Is $\Omega$ the set of all possible events?

solution

No, $\Omega$ is one particular event (the event containing everything). The set of all possible events is much larger!

What is the set of all possible events for Chibany’s situation?

solution

$\{ \{ \}, \{ HH \}, \{ HT\}, \{TH \}, \{TT\}, \{HH,HT\}, \{HH,TH\}, \{HH,TT\}, \{HT, TH\}, \{HT, TT \}, \{TH, TT\}, \{HH, HT, TH\}, \{HH, HT, TT \}, \{HH, TH, TT\}, \{HT, TH, TT\}, \{HH, HT, TH, TT\} \}$

Note that $\{ \}$ is called the empty or null set and is a special set that contains no elements. It’s the impossible event where nothing happens. Chibany would never allow himself to not get any meals!

Counting tip: For an outcome space with $n$ outcomes, there are $2^n$ possible events. Here: $2^4 = 16$ events.

What We’ve Learned

In this chapter, Chibany introduced us to the fundamental building blocks of probability:

Sets: Collections of distinct elements
Outcome spaces ($\Omega$): All possible outcomes
Events: Subsets of outcomes we’re interested in

Next, we’ll see how to turn these into actual probabilities!

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