MATH2209 Chap.2 Probability Rules and Bayes
Probability Rules and Bayes
Probability becomes easier when ordinary language is converted into events and reference groups. This chapter separates intersections, unions, complements, independence and mutual exclusivity before building conditional probability and Bayes' rule. The examples use trees to make denominators visible, so a probability such as condition given positive is not confused with positive given condition.
Every result is checked against its prior probability and the direction of the evidence.
What this chapter covers
- 01
Complements and unions
- 02
Intersections and overlap
- 03
Conditional probability
- 04
Independent events
- 05
Mutually exclusive events
- 06
Bayes' rule and reversed conditions
Worked example · free
Update risk after a positive result
- 1The joint factor-positive path is 0.30 x 0.80 = 0.24.
- 1The no-factor positive path is 0.70 x 0.10 = 0.07.
- 1All positive results have probability 0.24 + 0.07 = 0.31.
- 1Bayes gives 0.24 / 0.31 = 0.774.
- 1Interpret 77.4% among positive participants.
Key terms
- Conditional Probability
- Probability calculated within a group already known to satisfy another event.
- Independent Events
- Events for which learning one does not change the probability of the other.
- Bayes' Rule
- A rule that reverses a condition by combining prior probability with evidence likelihood.
Probability Rules and Bayes FAQ
Why is sensitivity not the answer after a positive result?
Sensitivity conditions on people who truly have the condition. A post-positive probability conditions on everyone who tested positive, including false positives. Because the denominator changes, Bayes' rule must include both positive paths rather than copy the sensitivity value.
Are mutually exclusive events independent?
Usually not. Mutually exclusive events cannot occur together, so observing one forces the probability of the other to zero. Independent events leave one another's probabilities unchanged. Only special zero-probability cases avoid that conflict.
How can I catch a reversed condition?
Write both quantities in words. Positive given disease asks how the test behaves within diseased participants; disease given positive asks how positive participants are composed. A tree or two-by-two table makes the two denominators visibly different.
What checks should follow a Bayes calculation?
Confirm that every branch leaving a node sums to one and that each joint path was formed by multiplication. The numerator must be one of the paths included in the denominator, so it cannot exceed the total evidence probability. Compare the posterior with the prior: evidence that is more likely under the target condition should move probability upward, while weak evidence should produce a smaller change.
Retain the false-positive or alternative path because it explains why a positive result is not certainty. Finally, state the denominator group in words, keep the observed counts or probabilities visible and verify that the answer lies between zero and one. Rebuilding the same result from a two-way table is a strong independent check on a probability tree.
Assessment move
Translate every problem into named events before using numbers. Start by writing the event definitions and the requested probability in notation, then restate the condition in ordinary language. The words after the vertical bar define the restricted denominator. Build a small collection of paired questions such as alarm given contamination and contamination given alarm.
Solve both from the same four counts and explain why exchanging the condition changes the reference group. This is the best defence against confusing sensitivity with a post-positive probability. For union questions, draw overlapping circles or list the intersection explicitly. Calculate the two marginal probabilities, subtract the overlap once and check that the union cannot be smaller than either marginal.
For complements, name the complete set before subtracting from one. For independence, compare a conditional probability with its marginal counterpart or compare a joint probability with the product of marginals; do not rely on the events sounding unrelated. Use probability trees for multistage problems.
Put conditional probabilities on branches, multiply along one complete path and add mutually exclusive paths that reach the same final event. Check that branches leaving a node sum to one. Convert one tree each week into a two-way table and back again so that you can choose the clearer representation under time pressure.
For Bayes questions, write the numerator path first, then enumerate every path to the observed evidence for the denominator. Keep fractions or full decimals until the last line. Compare the posterior with the prior and ask whether the direction makes sense given the likelihoods. End every answer by naming the conditioned group, not merely reporting a decimal.
A useful timed drill is to solve one problem three ways: tree, table and formula. The numerical result should agree, while the explanation should always identify numerator, denominator and whether the requested event is a union, intersection, complement or conditional probability.
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