Conditional Probability, Independence and Bayes' Theorem
Conditioning restricts the sample space. Independence is the case where it changes nothing, and Bayes' theorem reverses the conditioning, which is how evidence updates a belief.
What a learner can do afterwards
- Compute a conditional probability from a joint and a marginal
- Apply Bayes' theorem to a screening test and interpret the result
- Distinguish independent events from mutually exclusive ones
1 · Read
You narrow the world when you condition. P(A given B) equals P(A and B) divided by P(B). With joint 0.24 and P(B) 0.6, the conditional chance is 0.24 divided by 0.6, which is 0.4. Along a tree path you multiply: the chance of both equals the first chance times the conditional chance of the second.
Take a high-risk group where 10 percent carry a condition. A test catches 90 percent of cases and clears 90 percent of healthy people. True positives are 0.1 times 0.9, which is 0.09. False positives are 0.9 times 0.1, which is also 0.09. Total positives are 0.18, so Bayes gives 0.09 divided by 0.18, which is 0.5. Half of all positives are real cases.
Independence means learning changes nothing: P(A and B) equals P(A) times P(B), so P(A given B) equals P(A). Mutual exclusivity means both cannot happen, so P(A and B) equals 0. With positive chances a pair cannot be both, since 0 cannot equal a positive product.
Split lateness over weather: P(rain) 0.3, P(late given rain) 0.5, P(late given no rain) 0.2. The rain branch gives 0.3 times 0.5, which is 0.15. The dry branch gives 0.7 times 0.2, which is 0.14. Add the branches: 0.15 plus 0.14 is 0.29.
Read a positive test against the base rate. When the condition is rare, healthy people outnumber cases, so false positives can match true ones. That is why an accurate test for a rare condition still returns many false alarms.
Conditioning narrows the world, independence ignores the news, and Bayes reverses the question.
2 · Watch
Take it off screen
Where it sits
Learn first
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.