Before We Begin

Harry Drinks Liquid Luck

Half-Blood Prince. Harry swallows a single dose of Felix Felicis — liquid luck. Golden warmth spreads through him. Everything suddenly feels obvious.

Harry

"I need to go to Hagrid's."

Hermione

"No. It has to be Slughorn — that's the whole point."

Harry

"I just… have a good feeling about it."

What Actually Happens

He goes to Hagrid's anyway — and Slughorn is already there.

The Catch

Harry Was Never Guessing

Felix Felicis didn't give Harry a feeling.
It gave Harry a correct feeling — chemically guaranteed.

We don't get that potion.

Every day, we feel like we've just taken a swig of liquid luck.

Most of the time, we haven't.

The Monty Hall Problem!

Can You Trust Your Intuition?

Mudassir Shabbir
Department of Computer Science
LUMS University

Intuition Fails

Three Fast Visual Puzzles

Medical Test

Positive result looks huge, base rate is tiny.

Coin Streak

HHHHHH

Long streaks feel like they must reverse.

Birthday Match

Jan 12

Mar 03

Jul 19

Oct 24

Feb 08

Dec 30

May 22

Oct 24

Apr 15

Class size: 60

Small groups can still make collisions surprisingly likely.

Long Lecture Module

Medical Test: Base Rates and Bayes

Model Setup

Assume prevalence P(D) = 1%.

Sensitivity P(+|D) = 99%.

False-positive rate P(+|~D) = 5%.

Compute P(D|+) with Bayes, not intuition.

Bayes Calculation

P(+) = 0.99(0.01) + 0.05(0.99) = 0.0594

P(D|+) = 0.99(0.01) / 0.0594

P(D|+) = 1/6 = 16.7%

Even with a strong test, rare events can dominate the denominator.

Long Lecture Module

Coin Streaks: Independence, Not Memory

Key Fact

For a fair coin, each toss is independent: P(T next | HHHHHH) = P(T next) = 1/2.

Independence means past outcomes do not alter future bias.

Where Intuition Fails

Humans expect local balancing in short windows. Probability guarantees balancing only in long-run frequencies.

Streaks are not evidence of correction; they are natural patterns in randomness.

Long Lecture Module

Birthday Collision: Exact Probability

Complement Strategy

P(match) = 1 - P(no match)

P(no match for n people) = prod(k=0 to n-1) (365-k)/365

Computing the complement is easier than direct counting.

For n = 60

P(no match) approx 0.006

P(match) approx 0.994

So a shared birthday is extremely likely.

Monty Hall

Pick One Door

Watch carefully. First we reveal, then shuffle.

Monty Opens a Door

One Goat Is Revealed

Stay or Switch?

Run the animation.

Poll

Stay

Switch

The Trap

50%

Looks symmetric

Feels 50-50

But information is asymmetric

Heart of the Idea

Where Did the Probability Go?

Picked Door

1/3

Opened Goat Door

1/3

Remaining Door

1/3

Proof

Why Switching Is Better (Always)

Case Split by Your First Pick

Car picked first: probability = 1/3

Car not picked first: probability = 2/3

Host always opens a goat door and never opens the car.

Winning Rules

If first pick is car: stay wins, switch loses.

If first pick is goat: stay loses, switch wins.

P(stay wins) = 1/3, P(switch wins) = 2/3

Switching wins exactly when your first pick was wrong, and that happens 2 out of 3 times.

Long Lecture Module

Monty Hall Assumptions (Rigorous Form)

Axioms

1) Car location is uniformly random.

2) Player picks one door.

3) Host knows car location.

4) Host always opens a goat door not picked by player.

5) Host always offers switch.

Why Assumptions Matter

If host behavior changes (sometimes opens randomly or does not always offer switch), posterior probabilities change.

The famous 2/3 result is model-dependent and therefore testable.

Long Lecture Module

Monty Hall via Bayes Rule

Notation

C_i: car behind door i.

H_j: host opens door j.

Assume player picked door 1.

Posterior Comparison

P(C_1|H_3) prop P(H_3|C_1)P(C_1) = (1/2)(1/3)

P(C_2|H_3) prop P(H_3|C_2)P(C_2) = (1)(1/3)

Normalize: P(C_1|H_3)=1/3, P(C_2|H_3)=2/3

Try Again

Five Fast Games

Press Start to run 5 rounds.

Monte Carlo

1000+ Simulations

Games: 0

Stay

Switch

Let the room watch convergence happen in real time.

Generalization

The 100-Door Version

Pick 1 door. Monty opens 98 goat doors.

Still feels 50-50?

Intuition Pitfalls

Probability Is Full of Surprises

🎂

Birthday paradox

🎲

Gambler's fallacy

🧒

Boy or girl paradox

🃏

Card paradox

💰

Simpson's paradox

Long Lecture Module

Birthday Paradox: Why It Grows Fast

Number of pairs among n people is n(n-1)/2, so comparisons scale quadratically.

Around n = 23, P(match) already exceeds 50%.

The paradox is not about strange probability; it is about hidden combinatorial growth.

Long Lecture Module

Gambler's Fallacy: A Formal Contrast

False Claim

"After many heads, tails is due."

This assumes negative dependence that is absent.

Correct Claim

For IID tosses: P(T next | history) = 1/2.

History changes surprise, not probability.

Long Lecture Module

Boy or Girl Paradox: Information Granularity

Statement A

"At least one child is a boy."

Possible ordered families: BB, BG, GB -> P(BB)=1/3.

Statement B

"The older child is a boy."

Possible: BB, BG -> P(BB)=1/2.

Different condition, different sample space.

Long Lecture Module

Card Conditioning: Same Data, Different Query

Example deck query: from a two-card hand, given one card is an Ace, what is P(both are Aces)?

Condition descriptions such as "this specific shown card is an Ace" vs "at least one Ace exists" induce different posteriors.

Lesson: condition statements are operators on sample spaces.

Long Lecture Module

Simpson's Paradox: Aggregates Can Reverse

Within Groups

Treatment A can outperform B in every subgroup.

After Aggregation

Overall rates can reverse due to different weights.

Always inspect confounders before making causal claims.

60-Minute Recap

Unified Principle

Probability errors are usually sample-space errors.

Define events carefully. Condition precisely. Verify numerically.

Intuition is useful for ideas; mathematics is required for calibration.

Closing

Why Study Probability?

Mathematics doesn't replace intuition.
It trains it.

Welcome to Probability.

Revisit the opening question: now you want to analyze before guessing.

Next up: conditional probability and Bayes' rule.