Basic Probability

Why probability exists (motivation)

In many everyday situations you can’t predict the exact outcome, but you can still reason about how likely different outcomes are.

• •A coin flip: you can’t control heads vs tails.
• •A die roll: you can’t choose which face lands up.
• •Drawing a card from a shuffled deck: you don’t know which card you’ll get.

Probability gives a precise language for “chance.” In this node we focus on the simplest, most concrete version: classical probability.

The two objects you always start with

1) Sample space (usually written Ω)

• •Definition: the set of all possible outcomes of an experiment.
• •Example: for one coin flip, Ω = {H, T}.
• •Example: for one die roll, Ω = {1,2,3,4,5,6}.

2) Event (often written A)

• •Definition: an event is a subset of the sample space, so A ⊂ Ω.
• •Intuition: an event is a “condition you care about.”
• •Example: on a die roll, “roll an even number” is the event A = {2,4,6}.

Probability of an event (the key symbol)

We write P(A) for “the probability that event A happens.”

In this node, we restrict to situations where outcomes are equally likely (fair coin, fair die, well‑shuffled cards, etc.). In that setting, the probability of an event is:

• •“Favorable” means “in the event A.”
• •|A| means “the number of elements in A.”

A tiny reality check: probabilities are numbers between 0 and 1

• •If A is impossible, A = ∅, then |A| = 0 and P(A) = 0.
• •If A is certain, A = Ω, then |A| = |Ω| and P(A) = 1.
• •Most events land in between.

Where the interactive canvas fits

Use the canvas as a counting tool:

• •First, decide what Ω is.
• •Then, highlight which outcomes belong to A.
• •Finally, compute P(A) = highlighted / total.

This is not just a visualization—it’s the workflow you’ll use repeatedly.

Why being explicit about Ω matters

A very common beginner mistake is to compute “favorable / total” with the wrong “total.” The total is not “whatever feels like all outcomes”—it’s exactly the size of your sample space Ω.

So we slow down and build a habit:

1) Name the experiment (what is being done?)

2) Write Ω (all outcomes)

3) Describe A as a subset of Ω

4) Count |Ω| and |A|

Example 1: One coin flip

Experiment: flip a fair coin once.

• •Ω = {H, T}
• •Event A = “heads” = {H}
• •|Ω| = 2, |A| = 1
• •P(A) = 1/2

Example 2: One die roll

Experiment: roll a fair six‑sided die.

• •Ω = {1,2,3,4,5,6}
• •Event A = “roll at least 5” = {5,6}
• •|Ω| = 6, |A| = 2
• •P(A) = 2/6 = 1/3

Composite outcomes: ordered pairs

When you repeat an experiment, each outcome often becomes a tuple.

Experiment: flip a coin twice.

A good sample space is ordered outcomes:

• •Ω = {HH, HT, TH, TT}
• •|Ω| = 4

Event A = “exactly one head”

• •A = {HT, TH}
• •|A| = 2
• •P(A) = 2/4 = 1/2

Notice how order matters: HT ≠ TH as outcomes (even if they both represent “one head”).

Use the canvas: the grid idea

In the interactive canvas, two dice outcomes are naturally shown as a 6×6 grid.

• •Each cell corresponds to one ordered pair (d₁, d₂).
• •That grid is the sample space Ω.
• •If the dice are fair, every cell is equally likely.

So if you can count highlighted cells, you can compute probability.

Set operations (just enough to be useful)

Because events are sets, you can combine them.

• •Union: A ∪ B = outcomes in A or B (or both)
• •Intersection: A ∩ B = outcomes in A and B
• •Complement: Aᶜ = outcomes not in A (within Ω)

These lead to quick probability facts in the equally‑likely setting:

• •P(Aᶜ) = 1 − P(A)

And if A and B do not overlap (A ∩ B = ∅), then

• •P(A ∪ B) = P(A) + P(B)

You don’t need more set theory than that for this node—just recognize events can be built and combined.

Why counting is the engine

In classical probability, the hardest part is usually not “probability” itself—it’s counting.

You already know counting principles (addition and multiplication rules). Here’s how they plug into probability:

• •Multiplication rule often builds |Ω| for multi‑step experiments.
• •Addition rule often builds |A| when an event can happen in multiple distinct ways.

The classical probability formula

When outcomes are equally likely:

The word “equally likely” is doing a lot of work.

• •Fair coin: each outcome has probability 1/2.
• •Fair die: each face has probability 1/6.
• •Two fair dice: each ordered pair has probability 1/36.

If outcomes are not equally likely, you need more advanced tools (later nodes). For now, we deliberately stay in the equally‑likely world.

Worked motivation: why 7 is the most likely sum (return to the puzzle)

Experiment: roll two dice and sum them.

Step 1: sample space

• •Each die has 6 outcomes.
• •Ω = {(d₁, d₂) : d₁ ∈ {1,…,6}, d₂ ∈ {1,…,6}}
• •|Ω| = 6×6 = 36

Step 2: event for sum = 6

A₆ = {(d₁,d₂) : d₁+d₂=6}

List them:

• •(1,5), (2,4), (3,3), (4,2), (5,1) → 5 outcomes

So |A₆| = 5 and P(sum=6)=5/36.

Step 3: event for sum = 7

A₇ = {(d₁,d₂) : d₁+d₂=7}

List them:

• •(1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes

So |A₇| = 6 and P(sum=7)=6/36=1/6.

Conclusion: 7 is more likely than 6 because there are more ordered pairs that add to 7.

Use the canvas (explicit instruction)

On the interactive canvas:

1) Toggle to “Two dice (grid)” mode.

2) Click “Highlight sum = 7.” Count highlighted cells: 6.

3) Click “Highlight sum = 6.” Count highlighted cells: 5.

4) The fraction highlighted/36 is the probability.

This is the exact same favorable/total rule—just visual.

A compact table for common experiments

A note on “equally likely” (don’t skip this)

If a die is loaded, or a coin is biased, you cannot assume P(face) = 1/6.

Classical probability is best viewed as:

• •a starting model for fair random mechanisms, and
• •a training ground for thinking with Ω, events, and counting.

Later, you’ll generalize to probability models where outcomes have different weights.

1) Connection to random variables (next unlock)

A random variable is a rule that assigns a number to each outcome in Ω.

In the two‑dice example:

• •Outcome is (d₁,d₂) ∈ Ω.
• •Define X(d₁,d₂) = d₁ + d₂.

X is a random variable: it turns outcomes into numbers.

Why this matters:

• •Events can be written using X.
• •Example event: “sum is 7” is {X = 7}.

Then probability becomes:

That is the same favorable/total idea, just written in a way that scales.

2) Connection to conditional probability (next unlock)

Conditional probability asks: how does the probability change once you learn something?

Even in the equally‑likely world, learning information shrinks your sample space.

Example (two dice):

• •Original Ω has 36 equally likely outcomes.
• •Suppose you learn the first die is 6.
• •New sample space: Ω′ = {(6,d₂) : d₂ ∈ {1,…,6}}
• •|Ω′| = 6

Now ask: what is the probability the sum is 7 given die 1 is 6?

Event A = “sum is 7.” Within Ω′, only (6,1) works.

• •favorable = 1, total = 6 → probability = 1/6.

This is the intuition behind $P(A\mid B)$:

• •condition on B → restrict to outcomes where B is true
• •then compute favorable/total inside that restricted space

You will formalize this in the Conditional Probability node.

3) A practical habit you’ll reuse

Whenever you feel stuck, write:

• •What is Ω?
• •What is A?
• •Are outcomes equally likely?
• •Can I count |Ω| and |A|?

If you can answer those, you can compute P(A) in this basic setting.