STAT 400 Chap.2 Conditional Probability, Independence and Bayes Theorem
Conditional Probability, Independence and Bayes Theorem
Conditioning changes the reference set
Conditional probability answers a different question from an ordinary probability. P(A) measures A against the whole sample space; P(A | B) measures A only among outcomes where B happened. The defining ratio P(A∩B)/P(B) makes that change of reference explicit.
The course introduces the idea through a two-way table, where a marginal probability uses a row or column total and a conditional probability uses a single row or column as the new whole. Keeping that picture in mind prevents the most common confusion in the topic.
Trees turn the rule into a procedure
Rearranging the definition gives the multiplication rule, P(A∩B) = P(B)P(A | B).
A probability tree applies it repeatedly: first-level branches carry marginal probabilities, later branches carry conditional ones, and the probability of a whole path is the product along it. When the first-level branches form a partition, adding the paths that end in an event gives the law of total probability.
Most multi-stage questions in this chapter reduce to drawing the tree correctly and labelling each branch with the right kind of probability.
Independence is checked with numbers
Two events are independent when the probability of both equals the product of their probabilities. That definition has two consequences worth memorising.
Independence carries over to complements, which makes at least one questions easy through one minus the probability of none. And independence is not the same as mutual exclusivity: disjoint events with positive probabilities are always dependent.
For three or more events, every pair can be independent while the three together are not, so mutual independence needs the triple product as well.
Bayes theorem reverses the condition
Bayes theorem turns probabilities of evidence given a cause into probabilities of a cause given evidence.
In tree language, the posterior probability of one branch is its path probability divided by the total of all paths that lead to the evidence.
The lesson students remember from screening examples is the role of the base rate: a test that is accurate for individuals can still produce mostly false positives when the condition is rare, because the few true positives are outnumbered by false positives from the large healthy group.
Where it leads
Conditioning returns in joint distributions, where conditional pmfs are rows of a table divided by their totals, and in hypothesis testing, where a p-value is a probability computed under the condition that the null hypothesis holds.
What this chapter covers
- 01
Conditional probability as a ratio
- 02
Multiplication rule and probability trees
- 03
Law of total probability
- 04
Independence and its complements
- 05
Pairwise versus mutual independence
- 06
Bayes theorem and base rates
Worked example · free
Posterior probability from a two-stage tree
- 1Label the tree: P(W) = 0.60, P(V) = 0.40, P(pass | W) = 0.70, P(pass | V) = 0.55.
- 2Total probability: P(pass) = 0.60(0.70) + 0.40(0.55) = 0.42 + 0.22 = 0.64.
- 2Bayes: P(W | pass) = 0.42/0.64 = 0.656.
- 1Check: P(V | pass) = 0.22/0.64 = 0.344, and the two posteriors sum to 1.
Key terms
- Conditional Probability
- The probability of A computed only among outcomes where B occurred, P(A | B) = P(A∩B)/P(B) for P(B) > 0.
- Partition
- A collection of mutually exclusive events whose union is the whole sample space, so exactly one of them occurs.
- Law of Total Probability
- The rule that the probability of an event equals the sum, over a partition, of each part's probability times the conditional probability of the event given that part.
- Independent Events
- Events whose joint probability equals the product of their individual probabilities, so one carries no information about the other.
- Posterior Probability
- The updated probability of a cause after observing evidence, computed with Bayes theorem from the prior and the likelihood of the evidence.
- Base Rate
- The prior proportion of a condition in the population, which strongly controls how a positive test result should be read.
Conditional Probability, Independence and Bayes Theorem FAQ
What is the difference between independent and mutually exclusive events?
Mutually exclusive events cannot happen together, so their joint probability is zero. Independent events satisfy P(A∩B) = P(A)P(B). If both events have positive probability, being mutually exclusive forces dependence.
When should I use Bayes theorem instead of the definition of conditional probability?
Use Bayes when you are given probabilities of evidence under each cause but asked for a cause given the evidence. The definition still applies; Bayes simply rewrites its denominator with the law of total probability.
Why do screening tests give so many false positives?
When a condition is rare, the healthy group is much larger than the affected group, so even a small false positive rate produces more false alarms than the test produces true detections.
How do I show three events are mutually independent?
Check all three pairwise product rules and the triple product rule. Passing the pairwise checks alone is not enough, because the triple intersection can still fail the product rule.
Exam move
Draw every multi-stage problem as a tree before writing formulas, and label which branches are marginal and which are conditional. Rework one screening problem with natural frequencies, such as counts out of 10,000, to see the base-rate effect directly.
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