In This Lesson Probability Basics Conditional Probability & Bayes' Theorem 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 Discrete Distributions Continuous Distributions Central Limit Theorem 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 Probability Basics
P(A) = favorable outcomes / total outcomes
0 ≤ P(A) ≤ 1
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(A') = 1 − P(A)
Probability quantifies uncertainty. These rules come from set theory — unions and intersections. Counting techniques like combinatorics (permutations and combinations) are essential for computing probabilities in finite sample spaces.
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 Conditional Probability & Bayes' Theorem
P(A|B) = P(A ∩ B)/P(B)
Independent events: P(A ∩ B) = P(A)·P(B)
Bayes: P(A|B) = P(B|A)·P(A)/P(B)
Bayes' Theorem is the backbone of Bayesian inference and machine learning. It lets us update beliefs with new evidence. Total probability connects this to partition: P(B) = Σ P(B|Aᵢ)·P(Aᵢ).
Example: Disease Testing Disease prevalence: 1%. Test sensitivity: 99%. False positive rate: 5%.
P(Disease | Positive) = (0.99×0.01)/(0.99×0.01 + 0.05×0.99) ≈ 0.0099/0.0594 ≈ 16.7%
Even with a good test, a positive result is only 16.7% likely to be a true positive when the disease is rare!
Discrete Distributions Bernoulli: Single trial, p = success. E[X] = p, Var = p(1−p) Binomial(n, p): P(X = k) = C(n,k)·pᵏ(1−p)ⁿ⁻ᵏ — uses polynomial expansion Poisson(λ): P(X = k) = e⁻λ·λᵏ/k! — models rare events per interval Geometric(p): P(X = k) = (1−p)ᵏ⁻¹·p — trials until first success. Connects to exponential functions Continuous Distributions
Normal: f(x) = (1/σ√(2π))·e^(−(x−μ)²/(2σ²))
Z-score: z = (x − μ)/σ
Standard normal: μ = 0, σ = 1
For continuous distributions, probabilities are areas under the curve — you need integration. The normal (Gaussian) distribution is the most important, governing everything from measurement error to stock prices.
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 Uniform(a, b): f(x) = 1/(b−a) — constant 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 Exponential(λ): f(x) = λe⁻λˣ — time between events t-distribution: Used in hypothesis testing with small samples Chi-squared: Sum of squared standard normals — used in goodness-of-fit 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 Central Limit Theorem Central Limit Theorem: Regardless of the population distribution, the sample mean X̄ approaches a normal distribution N(μ, σ²/n) as n → ∞. This is why the normal distribution is so important, and why
statistical inference works. The convergence concept mirrors
limits in calculus .