Probability & Statistics
14 concepts
N = m × nCounting Cases
Things that can't happen together you ADD (the sum rule); things that happen one after another you MULTIPLY (the product rule).
P(A) = a / nWhat Probability Means
Count the equally likely outcomes, then take the share of them that are the ones you're hoping for.
P(Aᶜ) = 1 − P(A)Complementary Events
The probability that something does NOT happen is the whole probability 1 minus the probability that it does.
x̄ = (x₁ + x₂ + ⋯ + xₙ) / nMeasures of Center
The mean is the sum divided by the count, the median is the middle value in order, and the mode is the value that shows up most.
σ² = { (x₁−x̄)² + ⋯ + (xₙ−x̄)² } / nSpread: Variance & Standard Deviation
The average of the squared deviations of each value from the mean is the variance; its square root is the standard deviation.
ₙPᵣ = n!/(n−r)!, ₙCᵣ = n!/(r!(n−r)!)Permutations & Combinations
Picking r from n items: if the order you picked them in matters, it's a permutation (P); if only 'who got picked' matters, it's a combination (C).
P(B|A) = P(A∩B) / P(A)Conditional Probability
Given that event A has already happened, it's the probability B happens under that condition. You shrink the world to A, then weigh B.
P(X=k) = ₙCₖ pᵏ (1−p)ⁿ⁻ᵏThe Binomial Distribution
It's the distribution of the probability of getting k successes when you run n independent trials, each with the same success probability p.
P(μ−σ ≤ X ≤ μ+σ) ≈ 0.68The Normal Distribution
A bell-shaped distribution, symmetric about its mean. Data clusters most densely near the mean and thins out sharply as you move away.
E(X) = Σ xᵢ pᵢExpected Value & Variance
When you know the probabilities of the outcomes, it's each value times its probability, all summed. Repeat enormously many times and it's the 'long-run average' things settle to.
P(A|B) = P(B|A)·P(A) / P(B)Bayes' Theorem
Bayes' theorem is the rule for updating a belief with new evidence. Take the probability of A before seeing B (the prior), multiply by how likely B is if A holds, divide by the total chance of B, and you get the probability of A after seeing B (the posterior).
x̄ₙ = (X₁ + X₂ + ⋯ + Xₙ) / n → μ (n → ∞)The Law of Large Numbers
The law of large numbers says that if you repeat the same random experiment many times, the average of the results gets closer and closer to the theoretical expected value μ. Individual outcomes stay unpredictable, but the average becomes predictable.
(x̄ₙ − μ) / (σ/√n) → N(0, 1) (n → ∞)The Central Limit Theorem
The central limit theorem says that whatever the shape of the original distribution—flat or skewed—if you gather many independent samples and take their average (or sum), the distribution of those averages approaches the normal (bell) shape.
P(Xₙ₊₁ | X₀…Xₙ) = P(Xₙ₊₁ | Xₙ); πP = πMarkov Chains
It is a process that hops randomly among states, where the probability of the next state depends only on the current state and not at all on the path taken to get there. This 'memorylessness' is the Markov property.
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