STAT1008 Chap.6 Conditional Probability, Bayes and Counting
Conditional Probability, Bayes and Counting
Conditional Probability, Bayes and Counting develops the general multiplication rule, the marginal probability rule written through conditionals, and probability trees that multiply along paths and add across them. Bayes’ theorem then reverses a condition, updating prior probabilities after evidence is observed.
The chapter finishes with the counting rules used for classical probability: the multiplication rule, factorials, permutations when order matters and combinations when it does not. Assessment items in this topic tend to hide the structure in words: shares of a population, then rates within each share, then a question about the reverse condition.
Drawing the tree or a table of joint probabilities first turns the wording into arithmetic that a marker can follow line by line.
What this chapter covers
- 01
The general multiplication rule
- 02
Probability trees
- 03
The marginal probability rule through conditionals
- 04
Bayes’ theorem
- 05
Factorials and the multiplication rule for counting
- 06
Permutations and combinations
- 07
Using the complement for at-least questions
Worked example · free
Which supplier made the defective part?
- 1Priors: P(X) = 0.6 and P(Y) = 0.4. Likelihoods: P(D | X) = 0.03 and P(D | Y) = 0.08.
- 1Joint probabilities: 0.03 × 0.6 = 0.018 and 0.08 × 0.4 = 0.032.
- 1Marginal probability rule: P(D) = 0.018 + 0.032 = 0.0500.
- 2Bayes’ theorem: P(Y | D) = 0.032/0.0500 = 0.6400.
Key terms
- Prior probability
- A prior probability is the probability of a hypothesis before new evidence is taken into account.
- Permutation
- A permutation is an ordered arrangement of x objects chosen from n distinct objects.
- Combination
- A combination is a selection of x objects from n where the order of selection does not matter.
- Probability tree
- A probability tree is a branching diagram whose path products give joint probabilities for successive stages.
- Factorial
- The factorial n! is the product of the whole numbers from n down to 1, with 0! equal to 1.
- Posterior probability
- A posterior probability is the updated probability of a hypothesis after new evidence has been observed.
Conditional Probability, Bayes and Counting FAQ
How do I know whether a question needs Bayes’ theorem?
Look for a reversed condition. If the stem gives probabilities of evidence given each cause, and asks for the probability of a cause given the evidence, Bayes’ theorem is the tool. A probability tree makes the same calculation visible.
When does order matter in a counting problem?
Swap two of the chosen items and ask whether the outcome changes. A password or a finishing order changes, so use permutations. A committee or a hand of cards does not, so use combinations.
Why might nPr overcount an arrangement?
The permutation formula assumes every object is distinct. When some items are identical, such as repeated letters, different orders of identical items look the same, so split the count into cases or divide out the repeated orders.
What is the marginal probability rule?
When events B1 to Bk are mutually exclusive and collectively exhaustive, the probability of A is the sum of P(A given Bi) times P(Bi) over all i. In a tree it is the sum of every path that ends in A, and it supplies the denominator of Bayes’ theorem.
How do I count selections with a condition such as exactly two of one type?
Multiply the ways to choose each type separately. For exactly two of four specialists and one of three others, there are 4C2 times 3C1 selections. Divide by the total number of selections to turn the count into a probability.
Exam move
Solve each Bayes problem three ways, with a tree, a two-way table of joint probabilities and the formula, until you can choose the fastest under time pressure. For counting, write “order matters” or “order does not matter” before every calculation, and practise the nCr key sequence on the exam-style calculator. Use the complement whenever a question says “at least one”.
A short daily drill of three counting questions and one Bayes question keeps both skills fresh until the final exam. Build a habit of checking every Bayes answer for direction: the hypothesis that makes the evidence more likely should gain probability.
For counting, keep a list of solved patterns, such as at least one, exactly k of one type and arrangements with repeats, and match each new question to a pattern before calculating.
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