Last lesson’s definition of conditional probability — P(A | B) = P(A and B) / P(B) — rearranges into the workhorse of all probabilistic reasoning, the multiplication rule:
P(A and B) = P(A | B) · P(B)
Read it as a path: to reach the world where both A and B hold, first B has to happen (that costs P(B)), then A has to happen in the world where B already did (that costs P(A | B), the conditional — output depends on what you’ve already walked through).
Your incident history says:
- 30% of production incidents are config-related.
- When an incident is config-related, it’s detected within 10 minutes of the triggering deploy 80% of the time (config errors tend to blow up fast).
- Across all incidents, only 50% are detected within 10 minutes.
Compute the probability that the next incident is both config-related and detected within 10 minutes.
One deliberate wrinkle: you were given three numbers, and one of them is a decoy for this particular question. Part of using the multiplication rule is picking the factorization that matches the information you actually have — you know P(fast | config), not P(config | fast), so chain in the direction your data supports.