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Probability and statistics — Formula Sheet

9 formulas from Probability & Statistics, each with what it means, when to use it and the trap it sets.

(k n) = \frac{n !}{k ! ( n - k )!}, P (n, k) = \frac{n !}{( n - k )!}

What it says: Combinations ignore order; permutations count it.
When to reach for it: Counting outcomes before any probability can be computed.
Watch out: Ask whether rearranging the same items counts as a new outcome. If not, it is a combination.

(2 5) = 10

Ten possible pairs from five people, since a pair has no order.

P (A \cup B) = P (A) + P (B) - P (A \cap B)

What it says: Add the two chances, then remove the overlap you counted twice.
When to reach for it: Anything phrased with "or".
Watch out: The overlap term vanishes only for mutually exclusive events. Assuming it always does inflates the probability.

P = \frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \frac{2}{3}

The subtraction is what keeps the total below 1.

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}

What it says: The chance of A once you already know B happened.
When to reach for it: Anything phrased with "given that".
Watch out: P(A|B) and P(B|A) are different numbers, and confusing them is the base-rate fallacy.

P (A \cap B) = P (A ∣ B) P (B)

The same rule, rearranged into the multiplication rule.

P (A ∣ B) = \frac{P ( B ∣ A ) P ( A )}{P ( B )}

What it says: Turn a conditional probability around using the prior.
When to reach for it: Medical tests, spam filters, and any "given a positive result, what is the chance" question.
Watch out: The denominator usually needs the law of total probability: P(B) = P(B|A)P(A) + P(B|A′)P(A′). A rare condition keeps P(A|B) low even after a positive test.

P (B) = P (B ∣ A) P (A) + P (B ∣ A^{c}) P (A^{c})

Compute the denominator first — it is where the marks are lost.

E [X] = \sum x_{i} p_{i}, Var (X) = E [X^{2}] - (E [X])^{2}

What it says: The long-run average, and how far outcomes typically sit from it.
When to reach for it: Summarising any distribution, discrete or continuous.
Watch out: Variance is E[X²] minus the square of the mean, in that order. Reversing it gives a negative number.

Var (X) = E [(X - μ)^{2}]

The equivalent definition, usually slower to compute.

P (X = k) = (k n) p^{k} (1 - p)^{n - k}

What it says: Exactly k successes in n independent tries, each succeeding with probability p.
When to reach for it: A fixed number of independent yes/no trials with a constant success rate.
Watch out: All four conditions must hold. Sampling without replacement breaks independence and needs the hypergeometric instead.

E [X] = n p, Var (X) = n p (1 - p)

Worth memorising alongside the probability itself.

z = \frac{x - μ}{σ}

What it says: How many standard deviations x sits above or below the mean.
When to reach for it: Normal probabilities, and comparing values from different distributions.
Watch out: For a sample mean the denominator is σ/√n, not σ — that √n is the whole point of the central limit theorem.

z = \frac{x - μ}{σ / n}

The sampling-distribution version, used for intervals and tests.

x ˉ \pm z^{*} \frac{σ}{n}

What it says: The sample mean, plus and minus a margin that shrinks as the sample grows.
When to reach for it: Estimating a population mean from a sample.
Watch out: Quadrupling n only halves the width — √n, not n. And 95% describes the procedure, not the probability that this one interval is right.

z^{*} = 1.96 for 95%

Use t* instead when σ is unknown and n is small.

b = r \frac{s _{y}}{s _{x}}, a = y ˉ - b x ˉ

What it says: The line minimising total squared vertical distance to the data.
When to reach for it: Fitting a straight line to a scatter of points.
Watch out: The line always passes through (x̄, ȳ), and correlation is not causation however good r looks.

r^{2} = fraction of variation explained

r² of 0.8 means 80% of the variation is accounted for by the line.

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