Conditional Probability & Distributions

Conditional probability asks: given that B happened, how likely is A?

P(A|B) = P(A and B) / P(B)

Knowing B happened shrinks the world to just the B outcomes, so you divide by P(B) to rescale.

If knowing B tells you nothing about A, the events are independent and P(A|B) = P(A).

The normal distribution is the bell curve that describes heights, measurement errors and test scores. It is symmetric about the mean, and the empirical rule says roughly:

• 68% of values fall within 1 standard deviation of the mean
• 95% within 2
• 99.7% within 3

So a score 2 standard deviations above average beats about 97.5% of the group.

Conditional probability is defined as P(A|B) = P(A ∩ B)/P(B) for P(B) > 0, which rescales the sample space to B. Rearranging gives the multiplication rule P(A ∩ B) = P(B)P(A|B), and independence is the condition P(A|B) = P(A), equivalently P(A ∩ B) = P(A)P(B).

Bayes' theorem, P(A|B) = P(B|A)P(A)/P(B), inverts the conditioning and is the basis of diagnostic reasoning. Its most-cited consequence is that a test with high sensitivity can still yield mostly false positives when the base rate P(A) is low — an outcome that is counterintuitive but follows directly from the arithmetic.

The normal distribution N(μ, σ²) is symmetric and unimodal, with the empirical rule giving approximately 68/95/99.7% within one, two and three standard deviations. Standardising via z = (x − μ)/σ converts any normal variable to the standard normal, so a single table serves all cases.

The Central Limit Theorem explains the distribution's ubiquity: sample means approach normality as sample size grows regardless of the underlying population's shape, which is what licenses normal-based inference throughout statistics.