Statistics: Probability & Distributions
Statistics lets us understand data, measure uncertainty, and make predictions. Probability theory gives us the mathematical foundation for randomness and inference.
Key Concepts
Probability Basics
P(event) = favorable outcomes / total outcomes. P ranges from 0 (impossible) to 1 (certain). Complement rule: P(not A) = 1 − P(A). P(A and B) = P(A) × P(B) if independent.
Permutations & Combinations
Permutations count ordered arrangements: P(n,r) = n!/(n−r)!. Combinations count unordered selections: C(n,r) = n!/(r!(n−r)!). Use P when order matters, C when it doesn't.
Normal Distribution
Bell-shaped curve, symmetric around the mean μ. Standard deviation σ measures spread. The 68-95-99.7 rule: 68% of data within 1σ, 95% within 2σ, 99.7% within 3σ.
Statistical Inference
A sample statistic (x̄, mean) estimates a population parameter (μ). Larger samples give more reliable estimates. Hypothesis testing checks if results are due to chance.
Live Python Practice
Interactive Lab
Check Your Understanding
A bag has 4 red and 6 blue marbles. P(red) =
C(5,2) — choosing 2 from 5 — equals:
In a normal distribution, about 95% of data falls within: