Multiplication Rule (Independent Events)

Independence means learning that A happened tells you nothing about B, and then the joint chance is a plain product: two fair coins both landing heads is 0.5 × 0.5 = 0.25. Because probabilities are fractions, multiplying shrinks them fast: a component that fails 1 time in 100 gives a five-component chain a 0.99⁵ ≈ 0.951 chance of surviving, so reliability engineers watch products, not sums. Enter each probability as a decimal from 0 to 1, or switch the unit to %.

On 18 August 1913 the roulette wheel at Monte Carlo landed on black twenty-six times running, and gamblers lost fortunes betting ever larger sums on red, convinced it was "due". The wheel had no memory: each spin remained an independent 18/37 ≈ 0.486. That is the trap in both directions: assuming independence where it does not exist is just as costly. Two mortgages defaulting are not independent when the same recession hits both borrowers, which is roughly how the models of 2007 went wrong.