CDT502 Chap.3 Probability and Conditional Evidence
Probability and Conditional Evidence
This chapter treats probability as a language for events and changing information. It drills complement, addition and multiplication rules, distinguishes mutually exclusive from independent events, and makes conditional direction visible through trees and imagined cohorts. Bayes reasoning is tied to screening decisions so that sensitivity is not mistaken for the probability of a condition among flagged cases.
Begin every exercise by defining the event and relevant cases in words. The complement covers everything outside one event. A union collects cases in either event and subtracts overlap that would otherwise be counted twice. An intersection requires both conditions. Those meanings decide the calculation before symbols are introduced and prevent a remembered formula from being applied to the wrong event structure.
Independence concerns information: observing one event leaves the chance of the other unchanged. Mutual exclusivity concerns overlap: the events cannot happen together. The distinction becomes clear through a concrete process. Outcomes on the same single draw may be mutually exclusive, while results from separate well-controlled draws may be independent. Never infer either property merely from labels that sound unrelated.
Conditional probability changes the denominator. Reading P(A given B) aloud as the share of B cases that also have A makes reversal visible. An imagined cohort then turns rates into counts, preserving every route to the evidence. This is particularly valuable in screening, where a rare condition can produce many more false flags than true flags even when sensitivity sounds impressive.
Bayes reasoning updates belief; it does not set the action threshold. A reversible manual review may be justified at a posterior probability far below the level needed for an irreversible penalty. State the costs of false positives and false negatives, capacity and available follow-up evidence after calculating. That final separation keeps a probability model from becoming an automatic policy.
What this chapter covers
- 01
Objective and subjective probability
- 02
Complements, unions and intersections
- 03
Mutual exclusivity and independence
- 04
Conditional probability
- 05
Probability trees
- 06
Bayes theorem and base rates
Worked example · free
Update a screening flag with the base rate
- 2Split an imagined cohort by the ten per cent prior.
- 3Apply each conditional flag rate within its branch.
- 3Divide true review flags by every flag and interpret.
Key terms
- Independent Events
- Events for which learning that one occurred does not change the probability of the other.
- Mutually Exclusive Events
- Events that cannot occur together, making their intersection empty.
- Posterior Probability
- The probability of a condition after incorporating specified evidence.
Probability and Conditional Evidence FAQ
What is the difference between independent and mutually exclusive?
Mutually exclusive events cannot occur together, so their intersection is zero. Independent events may occur together, but observing one does not change the probability of the other. Nontrivial mutually exclusive events are therefore not independent because observing one makes the other impossible.
Why should I turn percentages into an imagined cohort?
A cohort exposes every denominator and route to the evidence. Choose a convenient total, split by the prior, apply conditional rates within each branch and count the complete evidence group. This prevents the common reversal between the probability of a flag given a condition and the condition given a flag.
Does a high posterior probability determine the action?
Not by itself. A decision threshold also depends on false-positive and false-negative costs, review capacity, reversibility and available alternatives. Probability updates belief; the loss structure turns that belief into action. State both rather than presenting a score as an automatic decision.
Exam move
Translate every symbol into a sentence. Mark or as a union, and as an intersection and given as a restricted denominator. Draw a complete tree with sibling branches summing to one, multiply along paths and add across paths leading to the same evidence. Finish each result by naming the decision threshold separately.
Practise with imagined cohorts of one hundred or one thousand until every rate becomes a count without hesitation. Then change only the base rate and predict the posterior direction while keeping sensitivity and false-positive rate fixed. Build paired cards for mutually exclusive versus independent and for likelihood versus posterior probability, using a concrete mechanism on each side.
When checking an answer, trace its denominator back to the branch or evidence group in the diagram. Finally, write two actions for the same posterior under different error costs. This separates probability updating from the policy threshold and prevents a numerical score from becoming an automatic decision.
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