Stop 02 · Math & Statistics for AITopics · 7/13

Probability Fundamentals

In short

A classifier saying '92% spam' is speaking probability. Before you can trust, tune or explain model outputs, you need the basic rules of chance.

Random variables, events, and basic probability rules — the language models use to express uncertainty in predictions.

The purple blob stands on a cricket pitch and flicks a coin into the air.

Think of it like this

The toss before a match

Before every cricket match, a coin decides who bats first. Nobody knows the next toss, but over a whole season, heads comes up about half the time. Probability is that long-run share. One toss is unpredictable, thousands of tosses are not.

Watch it happen

Ten tosses, counting heads

1/4After 1 toss: 1 head. Share of heads = 1/1 = 1.0. Way off 0.5, but it's only one toss.

The big ideas

What you need to know

Between 0 and 1

P = 0 means impossible, P = 1 means certain. All outcomes of one event add up to 1.

Complement rule

P(not A) = 1 − P(A). If rain has a 0.3 chance, no rain has 0.7.

Or means add (if they can't overlap)

Rolling a 1 or a 2: 1/6 + 1/6 = 1/3. If events can overlap, subtract the overlap once.

And means multiply (if independent)

Two sixes in a row: 1/6 × 1/6 = 1/36. Only valid when one result doesn't affect the other.

See it in code

Walk through the code

python

1import random2random.seed(42)3 4rolls = [random.randint(1, 6) for _ in range(60_000)]5print(rolls.count(6) / len(rolls))          # close to 1/6, about 0.1676 7even = sum(r % 2 == 0 for r in rolls) / len(rolls)8print(even)                                  # close to 0.59 10pairs = [(random.randint(1, 6), random.randint(1, 6)) for _ in range(36_000)]11both_six = sum(a == 6 and b == 6 for a, b in pairs) / len(pairs)12print(both_six, 1 / 36)                      # both near 0.028

1/4Fixing the seed makes the random numbers repeat each run, so results are reproducible.

Simulation is a great way to check a probability rule. If your formula and the simulation disagree, one of them is wrong.

Save yourself the pain

Common mistakes

Oops

Thinking after five heads in a row, tails is 'due'.

Do this instead

Each fair toss is independent. The coin has no memory, so the next toss is still 50-50.

Oops

Multiplying probabilities of events that affect each other.

Do this instead

Only multiply directly when events are independent. Otherwise use conditional probability: P(A and B) = P(A) × P(B given A).

Your mission

Check the rules by simulation

~15 min

Quick check

A model says P(spam) = 0.85 for an email. What is P(not spam)?

Go deeper

1Review basic probability rules: complement, union, intersection, conditional probability.
2Practice computing probabilities on simple discrete examples (dice, cards).
3Learn independence vs. conditional dependence between events.