Lecture 20: Probability Basics

The Hook

You flip a fair coin 5 times and get: H H H H H

What's the probability the next flip is heads?

Most people feel it "should" be tails - the coin is "due."

Answer: Still 50%. The coin has no memory.

This is the gambler's fallacy - one of many probability misconceptions we'll confront today.

Common Misconceptions

Before we build the mathematics, let's name what we're fighting:

Misconception	The Error	The Truth
Gambler's fallacy	Past outcomes affect future independent events	Independent events have no memory
Hot hand fallacy	Streaks indicate changed probability	Random processes produce streaks naturally
Law of averages (wrong)	Things "even out" in the short run	Convergence only happens in the long run
Base rate neglect	Ignoring prior probabilities	Must account for how common something is
Confusion of inverse	P(A\|B) = P(B\|A)	These are generally different!

This lecture will make these errors impossible.

Interactive: The Gambler's Fallacy

Coin Flip Simulator - Law of Large Numbers

Heads

Tails

50%

Heads %

Longest Streak

Number of flips: 100

Streaks are normal in random data! A streak of 5+ heads doesn't mean the coin is biased or that tails is "due." The proportion converges to 50% over many trials, but the absolute difference can keep growing.

Recognition: The SPREAD Primitive

SPREAD: "Distributed across possibilities"

When outcomes are uncertain, we describe them by how probability is spread across the possibilities.

Chemistry Context	What's Spread
Boltzmann distribution	Probability across energy states
Orbital shapes	Electron probability in space
Reaction outcomes	Products from competing pathways
Measurement uncertainty	Error across possible values
Molecular speeds	Probability across velocities

Sample Spaces and Events

Experiment

A process with uncertain outcome (flip coin, measure bond length, observe reaction)

Sample Space (Ω)

The set of ALL possible outcomes

Event

A subset of the sample space (a collection of outcomes)

Example: Roll a Die

Sample space: Ω = {1, 2, 3, 4, 5, 6}

Events:

A = "roll even" = {2, 4, 6}
B = "roll > 4" = {5, 6}
A ∩ B = "even AND > 4" = {6}
A ∪ B = "even OR > 4" = {2, 4, 5, 6}

Probability: The Axioms

Kolmogorov's Axioms

For any event A in sample space Ω:

Axiom 1: $P(A) \geq 0$ (probabilities are non-negative)

Axiom 2: $P(\Omega) = 1$ (something happens)

Axiom 3: For mutually exclusive events: $P(A_1 \cup A_2 \cup \cdots) = P(A_1) + P(A_2) + \cdots$

Everything else follows from these three axioms.

Immediate Consequences

$P(\emptyset) = 0$ (impossible event has probability 0)
$P(A^c) = 1 - P(A)$ (complement rule)
$P(A) \leq 1$ (probabilities bounded by 1)
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (inclusion-exclusion)

Independence

Independent Events

Events A and B are independent if: $P(A \cap B) = P(A) \cdot P(B)$

Equivalently: knowing B occurred doesn't change the probability of A.

Independence is a property of how events relate, not of the events themselves.

Two events from the same experiment can be dependent. Two events from different experiments are typically independent.

Example: Two Coin Flips (Independent)

A = "first flip is heads", B = "second flip is heads"

P(A) = 1/2, P(B) = 1/2

P(A ∩ B) = P(both heads) = 1/4 = P(A) × P(B) ✓

Example: Dependent Events

A = "roll is 6", B = "roll is even"

P(A) = 1/6, P(B) = 1/2

P(A ∩ B) = P(A) = 1/6 (if it's 6, it's automatically even)

Is 1/6 = (1/6)(1/2) = 1/12? No! These are dependent.

Independence ≠ Mutual Exclusivity

Mutually exclusive events are highly dependent - if A happens, B definitely doesn't!

Conditional Probability

The probability of A given that B has occurred:

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

We restrict our sample space to B and ask: what fraction is also in A?

Example: Die Roll

P(roll is 6 | roll is even) = P({6}) / P({2,4,6}) = (1/6) / (1/2) = 1/3

Given that it's even, 6 is one of three equally likely outcomes.

The Multiplication Rule

$$P(A \cap B) = P(A|B) \cdot P(B) = P(B|A) \cdot P(A)$$

The Confusion of the Inverse: P(A|B) ≠ P(B|A) in general!

P(wet sidewalk | rain) ≈ 1.0
P(rain | wet sidewalk) ≈ 0.3 (could be sprinklers!)

Bayes' Theorem

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$

This lets us "reverse" conditional probabilities.

Interactive: Medical Testing & Base Rate Neglect

Bayes' Theorem in Action

True Positives

False Positives

P(disease|+)

Disease Prevalence (%): 1%

Test Sensitivity (%): 95%

Test Specificity (%): 95%

With a 1% prevalence and 95% accuracy, a positive test only means ~16% chance of disease!

Most positive tests are false positives because the disease is rare.

Chemistry Applications

Reaction Branching

A molecule can react via two pathways with rate constants k₁ and k₂:

$$P(\text{pathway } i) = \frac{k_i}{k_1 + k_2 + \cdots}$$

Reaction Pathway Branching

k₁ (s⁻¹): 50

k₂ (s⁻¹): 30

k₃ (s⁻¹): 20

Quantum Measurement

In quantum mechanics, |ψ|² gives probability density:

$$P(a \leq x \leq b) = \int_a^b |\psi(x)|^2 \, dx$$

The wavefunction encodes how probability is spread across positions.

What "Random" Actually Looks Like

"Random data should look uniform, evenly spread."

Random data has clusters, gaps, and apparent patterns. If you see perfectly uniform spacing, that's evidence of NON-randomness!

100 Random Points - Notice the Clusters!

Summary: Probability Rules

Rule	Formula	Condition
Complement	P(Aᶜ) = 1 - P(A)	Always
Addition	P(A ∪ B) = P(A) + P(B) - P(A ∩ B)	Always
Addition (exclusive)	P(A ∪ B) = P(A) + P(B)	If A ∩ B = ∅
Multiplication	P(A ∩ B) = P(A\|B)P(B)	Always
Multiplication (indep.)	P(A ∩ B) = P(A)P(B)	If independent
Bayes	P(A\|B) = P(B\|A)P(A)/P(B)	Always

Misconception Remedies

If you think...	Remember...
"It's due to happen"	Independent events have no memory
"P(A\|B) = P(B\|A)"	Bayes' theorem shows they differ
"P = 0.9 means certain"	It means 9/10 times, not THIS time
"Random = uniform"	Random processes have clusters