The Uncertain Universe
Probability, Distributions & Expectation
The Human Problem
World 15 taught us how to count possibilities. World 24 taught us how to archive actual data. Now we ask the missing question: How do we mathematically measure uncertainty about what may happen next? We cannot predict the exact outcome, but we can assign a number to its likelihood. Probability turns uncertainty into a mathematical object.
RANDOMNESS →
EXPERIMENT →
OUTCOME →
SAMPLE SPACE →
EVENT →
COUNTING →
PROBABILITY →
CONDITIONAL →
INDEPENDENCE →
TOTAL PROBABILITY →
BAYES →
RANDOM VARIABLE →
DISTRIBUTION →
EXPECTATION
The 73 Probability Cells
From Coin Toss to Bayesian Inference
CELL 01-04
Randomness & Space
H,H
H,T
T,H
T,T
- Random Experiment
- Outcome (Single result)
- Sample Space (All results)
- Uncertainty before execution
CELL 05-12
Events
- Event as Subset (A ⊆ S)
- Simple vs Compound Events
- Certain (S) & Impossible (∅)
- Complement (Aᶜ)
- Mutually Exclusive (A ∩ B = ∅)
- Exhaustive Events
CELL 13-20
Classical Probability
- P(A) = |A| / |S|
- Probability Scale [0, 1]
- Empirical vs Long-Run
- Probability Axioms
- Union Rule (Inclusion-Exclusion)
- Equally Likely Assumption
CELL 21-24
Counting Connection
- Counting + Probability (World 15)
- Tree Diagrams
- Multiplication Along Paths
- Complement Counting
CELL 25-29
Conditional Probability
- P(A|B) = P(A ∩ B) / P(B)
- Shrinking the Sample Space
- Multiplication Theorem
- Conditional vs Ordinary Prob
CELL 30-33
Independence
- P(A|B) = P(A)
- P(A ∩ B) = P(A)P(B)
- Mutually Exclusive ≠ Independent
- Multiple-Event Independence
CELL 34-41
Total Prob & Bayes
P(A|B) ↔ P(B|A)
- Partition of Sample Space
- Law of Total Probability
- Bayes' Theorem
- Reverse Probability Reasoning
- Base Rate & Bayesian Updating
CELL 42-50
Random Variables
Outcome → Number
- Outcome → Numerical Value
- Discrete Random Variable
- Probability Mass Function
- Expectation (Weighted Mean)
- Linearity of Expectation
- Variance & Standard Deviation
CELL 51-59
Distributions
- Binomial (nCr p^r q^(n-r))
- Poisson Distribution
- Normal Distribution (Bell Curve)
- Standard Normal (Z-score)
- Continuous Variables & Density
CELL 60-69
Applications & Frontier
- "At Least" / "At Most"
- Risk vs Probability
- Physics, Bio, Finance, Computing
- Bayesian Networks (Frontier)
- Probability Models vs Reality
CELL 70-73
The Uncertainty Engine
COUNTING + DATA
↓
PROBABILITY
↓
INFERENCE & DECISION
↓
PROBABILITY
↓
INFERENCE & DECISION
- Counting → Probability
- Data → Probability Model
- Expectation → Decision
- Mastery of Uncertainty
Exit Portal
We have mapped the space of all possibilities and learned to assign numbers to uncertainty. But when many possible answers exist, how do we determine which choices are allowed—and which allowed choice is best? The next door moves us from uncertainty to optimization.