Probability & Statistics

1. Classical Probability

• For equally likely outcomes: P(A) = (number of favorable outcomes) / (total number of possible outcomes).
• Combinations (order doesn't matter): C(n, k) = n! / [k!(n−k)!] — used for "choosing k items from n."
• Permutations (order matters): P(n, k) = n! / (n−k)!.

Common mistakes: using permutations when the problem doesn't care about order (or vice versa); when drawing multiple items "without replacement," forgetting the total number of outcomes shrinks after each draw (e.g., choosing 2 red balls from a bag uses C(4,2)/C(10,2), not (4/10)²).

2. Conditional Probability and Independence

• Conditional probability: P(A|B) = P(A∩B) / P(B) — the probability of A, given that B has occurred.
• Independence: A and B are independent if P(A∩B) = P(A)·P(B) (equivalently, P(A|B) = P(A)).
• Bayes-style problems (e.g., "given the outcome, what's the probability of the cause"): break the population into groups, compute P(each group ∩ outcome), then divide by the sum of all such joint probabilities.

Common mistakes: confusing P(A|B) with P(B|A) — these are generally NOT equal; when solving a "reverse" conditional probability problem, forgetting to sum over ALL groups in the denominator, not just the one being asked about.

3. Mean, Variance, and Standard Deviation

• Mean (average): x̄ = (sum of all values) / (number of values).
• Variance: average of the squared deviations from the mean: σ² = (1/n)·Σ(xᵢ − x̄)².
• Standard deviation: σ = √(variance) — same units as the original data (variance's units are squared, which is why standard deviation is often more interpretable).

Common mistakes: forgetting to square the deviations before averaging (which would just give zero, since deviations cancel out); reporting variance when the question asks for standard deviation (or vice versa) — remember variance is in squared units.

4. Normal Distribution (Basics)

• The normal distribution is a symmetric, bell-shaped curve centered at the mean μ, with spread determined by the standard deviation σ.
• Empirical rule (68-95-99.7 rule): approximately 68% of data falls within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.

Common mistakes: assuming the empirical rule percentages apply to ANY distribution — they are specific to the normal distribution; misreading "within 2 standard deviations" as meaning just above OR just below (it means the symmetric range on both sides).