MZB127 Chap.7 Conditional Probability and Bayes' Rule
Conditional Probability and Bayes' Rule
Conditional probability restricts the reference population to the event after the conditioning bar. Intersections, unions and complements then support independence checks, total probability and Bayes' theorem. The chapter uses area diagrams, trees and imagined cohorts to keep denominators visible. Its central distinction is between a likelihood such as P(evidence given cause) and a posterior such as P(cause given evidence).
The event after the bar supplies the denominator, so reversing a condition usually changes the numerical answer even when the intersection is unchanged. Independence means that learning one event leaves the probability of the other unchanged; it can also be checked through a product rule for the intersection. Disjointness instead concerns an empty overlap and should not be confused with independence.
Total probability adds mutually exclusive joint paths that exhaust the observed evidence. Bayes' theorem then normalises one of those paths by the full evidence probability. Two-way tables, area diagrams, trees and imagined cohorts display the same relationships in different forms. Row totals, column totals, branch sums and posterior sums provide built-in checks.
The objective is not merely to remember a formula but to identify the reference population and preserve event direction through every step.
What this chapter covers
- 01
Events, complements and intersections
- 02
Conditional probability denominators
- 03
Independence tests
- 04
Total probability over a partition
- 05
Bayes updates and posterior checks
Worked example · free
Update a line source after a defect
- 1Form the A-to-defect joint path.
- 1Form the B-to-defect joint path.
- 1Add both paths for total defect probability.
- 1Divide the B path by that total.
- 1Compare the posterior with the prior share.
Key terms
- Conditional probability
- The probability of one event within the restricted reference space formed by another event.
- Independence
- A relationship in which learning one event does not change the probability of the other.
- Partition
- Mutually exclusive events that together cover the entire sample space.
- Likelihood
- The probability of observed evidence under a stated category or parameter value.
- Posterior probability
- The updated probability of a category after the evidence is observed.
Conditional Probability and Bayes' Rule FAQ
Why does reversing the condition change the answer?
The intersection may be the same, but the denominator changes. P(A given B) divides by P(B), while P(B given A) divides by P(A), so the two questions refer to different restricted populations.
Are disjoint events independent?
Not when both have positive probability. Disjoint events cannot occur together, so observing one forces the probability of the other to zero rather than leaving it unchanged.
What is the safest Bayes workflow?
List prior categories, multiply along every category-to-evidence path, add those paths to obtain the evidence probability, and divide the target path by the evidence total. Check that all posterior categories sum to one.
How does an imagined cohort check Bayes arithmetic?
Choose a cohort size that turns the priors and conditional rates into convenient counts. Count the evidence cases within every cause, add them, and ask what fraction came from the target cause. The resulting posterior should match the probability calculation, provided no intermediate count was distorted by rounding.
Exam move
Translate the conditioning bar into an “among” sentence and mark the denominator before calculating. Build a two-way table or tree when several conditional rates appear. Distinguish prior, likelihood, evidence and posterior in separate lines. For independence, verify either the intersection product or unchanged conditional probability.
Use an imagined cohort to audit a Bayes answer, but retain exact probabilities when rounding counts would distort the result. Start revision with verbal translations: for every expression, say which group forms the restricted population and which members count as successes. Build complete two-way tables from partial information and check every margin against the grand total.
Practise the same dataset in table, tree and probability notation so the joint cells and branch products become recognisably identical. When testing independence, write both the numerical equality and one sentence about whether observing an event changes the other probability. Keep disjointness in a separate comparison based on overlap.
For Bayes problems, label four quantities before arithmetic: prior cause share, evidence likelihood within each cause, total evidence and desired posterior. Multiply along branches, add across all evidence-producing branches and divide the desired joint path by that sum. Use a cohort size that converts rates cleanly when possible, but return to exact probability arithmetic if counts would require rounding.
Audit the result by checking that all posterior cause probabilities sum to one and by comparing the posterior with the prior in light of the likelihoods. In timed work, preserve the event labels on every numerator and denominator; unlabeled decimals make a reversed condition almost impossible to detect.
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