The formula
Conditional probability, the definition:
P(A | B) = P(A and B) / P(B), valid when P(B) > 0
Read it as: the probability of A once you already know B happened is the share of the B-outcomes that also carry A. You've thrown away every outcome where B didn't occur and re-scaled what's left so it sums to 1.
Rearrange it and you get the multiplication rule — the version you'll actually reach for at a bench:
P(A and B) = P(A | B) · P(B) = P(B | A) · P(A)
Read it as: two things happen together as often as the first happens times how often the second follows it.
Independence is a separate claim, and it has a clean test:
A and B independent ⇔ P(A and B) = P(A) · P(B)
⇔ P(A | B) = P(A)
Read it as: B carries no information about A. Learning B happened doesn't move the odds of A at all. The product form is the definition; the P(A|B) = P(A) form is what independence means in plain English.
Chain three or more events and it extends the obvious way:
P(A and B and C) = P(A) · P(B | A) · P(C | A and B)
For fully independent events the conditionals collapse and it's just the product of the individual probabilities.
Where you meet it
Reading a screening or diagnostic result at a receiving-inspection bench. A go/no-go tester lights up "reject." The number you actually want isn't the tester's accuracy — it's
P(part is bad | tester said bad). That's a conditional probability, and it depends hard on how many bad parts were in the lot to begin with. Skip the base rate and you'll scrap good hardware or ship junk.Redundancy math on a review board. You've got two "independent" channels — dual power supplies, A/B avionics strings, two pressure transducers voting. The whole safety case rests on multiplying two small failure probabilities. The board's job is to ask whether they're really independent, or whether one flood, one bad lot of capacitors, or one shared connector makes them fail together.
Fault isolation on a test stand. A transducer reads out of family.
P(sensor is the problem | reading is bad)versusP(the article is the problem | reading is bad)is exactly the conditional you're estimating in your head while you decide whether to swap the sensor or scrub the run.Reliability rollups. Series systems multiply survival probabilities; that multiplication is only legal when the components' failures are independent. Common environment — vibration, thermal cycling, one power bus — quietly breaks that assumption.
How it works
Conditioning is bookkeeping. P(A|B) restricts the sample space to B and renormalizes. Nothing more mystical than that. Two dice make the point cleanly: let A be "first die is even" and B be "the two dice sum to 7." Out of 36 equally likely rolls, P(A) = 18/36 = 1/2, P(B) = 6/36 = 1/6, and P(A and B) = 3/36 = 1/12. Since 1/2 · 1/6 = 1/12, these two are independent — even though both involve the same first die. Independence is a numerical coincidence in the probabilities, not a statement about physical connection. You can't eyeball it; you have to check the product.
The mistake people make, in order of how much it costs:
Confusing P(A|B) with P(B|A). These are not the same number and are often wildly different — this is the prosecutor's fallacy. Work a screening tester: prevalence of a real defect is 2%, the tester catches 95% of true defects (sensitivity) and correctly passes 90% of good parts (specificity, so a 10% false-alarm rate). When it says "reject":
P(reject) = 0.95·0.02 + 0.10·0.98 = 0.117 (11.7% of parts get flagged)
P(bad | reject) = 0.95·0.02 / 0.117 = 0.162 (about 16%)
A "95% accurate" tester, and roughly 84% of its rejects are good parts. Not because the tester is bad — because bad parts were rare to start with. The base rate dominates. Every engineer who's ever argued that a flagged unit is "probably" defective because the test is accurate has made this error.
Assuming independence that isn't there. This is the expensive one, because it hides in the exact place you're counting on it. Two channels each fail with probability q = 0.01. If truly independent, both fail with probability q² = 0.0001 — one in ten thousand. Now suppose 10% of each channel's failures come from a shared cause (same firmware bug, same vibration mode, same solder lot). The both-fail probability becomes roughly:
P(both fail) ≈ β·q + (1-β·q)·((1-β)q)² ≈ 0.00108
That's about ten times worse than the independent estimate — and it happened without changing either channel's individual reliability at all. Redundancy buys you far less than the multiplication suggests once a common cause is in the picture. This is why review boards chase shared power buses, shared connectors, shared calibration references, and shared design assumptions. Common-cause failure is the graveyard of paper redundancy.
Limits of validity worth keeping in front of you:
- Independent is not the same as mutually exclusive. Mutually exclusive events can't both happen, so
P(A and B) = 0— which makes them maximally dependent, not independent. People swap these words constantly. - Pairwise independence doesn't give you joint independence. Three events can be independent two-at-a-time and still be linked as a trio. If you need to multiply all three, pairwise checks aren't enough.
- Conditioning is undefined when
P(B) = 0. Fine for coins and dice; a real subtlety when B is a continuous measurement, which is what forces the measure-theoretic machinery underneath. - Independence is an assumption about the world, not a property you can prove from a data sheet. You justify it physically — separate power, separate lots, separate environments — or you don't get to use it.
History
The multiplication rule and the first clean statement of independence show up in Abraham de Moivre's The Doctrine of Chances, the first real textbook on probability, published in London in 1718 [1][2]. De Moivre (1667–1754) was a French Huguenot who fled to England, never landed a university chair, and made his living partly by computing odds for gamblers in London coffeehouses — which is exactly why his book is stuffed with dice and card problems. Buried in that gambler's manual is the idea that the probability of a compound event built from independent pieces is the product of their individual probabilities [1]. That product rule is the whole foundation of every redundancy calculation done since [2].
The conditional-probability machinery got its inverse-problem twist from Thomas Bayes and, more thoroughly, Pierre-Simon Laplace later in the century — turning P(evidence | cause) into P(cause | evidence), which is the diagnostic-reading problem above.
For nearly two centuries the definitions floated on intuition. That ended in 1933, when Andrey Kolmogorov published Grundbegriffe der Wahrscheinlichkeitsrechnung — Foundations of the Theory of Probability — and put the whole subject on measure-theoretic axioms [3][4]. In Kolmogorov's framework, P(A|B) = P(A and B)/P(B) and the product rule for independence stop being separate intuitions and become plain definitions inside a consistent system [4]. Everything downstream — conditional expectation, independence of random variables, the whole apparatus a reliability engineer leans on — follows from three short axioms. It's the reason your reliability spreadsheet and a pure mathematician's measure theory are describing the same object.
Related tools
- /tools/bearing-life-l10
- /tools/half-life-decay
- /tools/arrhenius-rate
- /tools/battery-life