BIOL2001 Chap.6 Probability Rules, Probability Trees and Bayes' Theorem
Probability Rules, Probability Trees and Bayes' Theorem
The rules behind every probability
Part II begins with the frequentist idea of probability, the proportion of times an event would occur in many repetitions, and the two properties that decide which rule to use. Mutually exclusive events cannot happen together, so their probabilities add; independent events give no information about each other, so their probabilities multiply.
When events overlap, the general addition rule subtracts the joint probability, and when they are dependent, the general multiplication rule uses a conditional probability.
Trees, tables and Bayes
A figure draws a probability tree for frogs from wet and dry habitat, multiplying along branches to find joint probabilities and adding across them to find the marginal probability of infection.
The same numbers fill a contingency table, test for independence and feed a worked Bayes' theorem calculation that reverses a conditional probability. The chapter also explains marginal, joint and conditional probability, independence in genetic inheritance, and why P(A | B) and P(B | A) differ.
What this chapter covers
- 01
Frequentist probability between 0 and 1
- 02
Mutually exclusive versus independent events
- 03
General addition rule
- 04
Multiplication rule and conditional probability
- 05
Probability trees
- 06
Contingency tables and marginal probabilities
- 07
Testing independence
- 08
Bayes' theorem
Worked example · free
Infection risk from a probability tree
- 1Multiply along the infected paths: 0.40 × 0.25 = 0.10 and 0.60 × 0.10 = 0.06.
- 1Add the paths for the marginal probability: P(infected) = 0.10 + 0.06 = 0.16.
- 1Apply Bayes: P(wet | infected) = 0.10 / 0.16 = 0.625.
- 1Under independence P(wet and infected) would be 0.40 × 0.16 = 0.064, not 0.10, so the events are dependent.
Key terms
- Mutually Exclusive Events
- Events that cannot happen at the same time, so their joint probability is zero.
- Independent Events
- Events for which knowing that one has happened tells you nothing about whether the other will.
- Joint Probability
- The probability that two events happen together.
- Marginal Probability
- The probability of an event regardless of the outcome of another variable.
- Bayes' Theorem
- The rule that reverses a conditional probability using P(B | A) = P(A | B) P(B) / P(A).
- Probability Tree
- A branching diagram whose paths multiply to joint probabilities and add to marginal ones.
- Complement Rule
- The rule that the probability an event does not occur is one minus its probability.
Probability Rules, Probability Trees and Bayes' Theorem FAQ
Are mutually exclusive events independent?
No. If two events cannot happen together, learning that one occurred tells you the other did not, which is the opposite of independence. Their joint probability is zero, so never multiply the probabilities of mutually exclusive events to find it.
How do I test whether two events are independent?
Compare the joint probability with the product of the two marginal probabilities. If P(A and B) equals P(A) times P(B), the events are independent; if not, they are dependent. The same check can be read from a contingency table of joint probabilities.
When should I use Bayes' theorem?
Use it when you know a conditional probability in one direction and need it in the other, such as the probability of infection given habitat when you want the probability of habitat given infection. Questions phrased as given that it tested positive usually call for it.
Why is the at least one shortcut useful?
The probability of at least one success equals one minus the probability of no successes. One subtraction replaces a sum over many outcomes, so for two eggs that each hatch with probability 0.8 the answer is one minus 0.2 squared, or 0.96.
Assessment move
For every probability question, first write whether it asks for or, and, or not, then pick the rule. Draw a tree for any two-stage process and check that the leaf probabilities add to one. Rebuild the frog example as a contingency table and read a conditional probability from it by dividing a cell by its margin.
Then swap the conditioning in one example on purpose and compute both P(A | B) and P(B | A), so that the difference between the two becomes something you have calculated rather than memorised.
Working through Probability Rules, Probability Trees and Bayes' Theorem in BIOL2001? Sia is AskSia’s AI Statistics tutor — ask any BIOL2001 Probability Rules, Probability Trees and Bayes' Theorem question and get a clear, step-by-step explanation grounded in how BIOL2001 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
STAT1008 Quantitative Research Methods · BIOL1030 Diversity of Life · STAT7038 Regression Modelling
