STAT1008 Chap.5 Probability Rules and Contingency Tables
Probability Rules and Contingency Tables
Probability Rules and Contingency Tables introduces random experiments, sample spaces and events, and the three sources of probability: a priori, empirical and subjective. It works through unions, intersections and complements on Venn diagrams, mutually exclusive and collectively exhaustive events, and marginal, joint and union probabilities read from a contingency table.
The chapter closes with conditional probability and the two equivalent tests the course uses for independence. The in-tutorial probability quiz uses one scenario with five parts, and the same structure suits exam questions. Most parts can be answered from a contingency table or a Venn diagram once events are defined, and the final part often asks whether two events are independent.
What this chapter covers
- 01
Random experiments, sample spaces and events
- 02
Three ways to assign a probability
- 03
Union, intersection and complement
- 04
Mutually exclusive and collectively exhaustive events
- 05
Probabilities from a contingency table
- 06
The general addition rule
- 07
Conditional probability and independence
Worked example · free
Probabilities from a table of 400 gym members
- 1Let A = annual member and B = attends classes. P(A) = 140/400 = 0.3500.
- 2P(A ∪ B) = 0.3500 + 0.4650 − 0.2100 = 0.6050 by the general addition rule.
- 1P(B | A) = 84/140 = 0.6000.
- 2P(B) = 0.4650 differs from P(B | A) = 0.6000, so the events are not independent in this sample.
Key terms
- Sample space
- The sample space is the collection of all simple outcomes a random experiment can produce.
- Mutually exclusive events
- Mutually exclusive events cannot occur together, so the probability of their intersection is zero.
- Conditional probability
- A conditional probability is the probability of one event given that another has occurred.
- Statistical independence
- Statistical independence holds when knowing one event occurred leaves the other event’s probability unchanged.
- Complement of an event
- The complement of an event contains every outcome in the sample space that is not in the event.
- Empirical probability
- An empirical probability is a relative frequency observed in data, used as an estimate of the true probability.
- Joint probability
- A joint probability measures the chance of two events happening together, written P(A and B).
Probability Rules and Contingency Tables FAQ
Are mutually exclusive events independent?
No, unless one has probability zero. If two events cannot happen together, learning that one occurred tells you the other did not, which changes its probability to zero. Independence requires the probability to stay the same.
Can I add two probabilities to get the chance of either event?
Only when the events are mutually exclusive. Otherwise use the general addition rule, which subtracts the probability of both occurring so the overlap is not counted twice. Without the joint probability the union cannot be found.
How do I test independence from a contingency table?
Compare a conditional probability with the matching marginal, such as P(B given A) with P(B), or compare P(A and B) with P(A) times P(B). For variables with several levels, the conditional distribution must match at every level.
What is the difference between joint and conditional probability?
A joint probability, such as P(A and B), is a share of everyone in the sample space. A conditional probability, such as P(A given B), is a share of only those in B, so its denominator is the size of B rather than the whole table.
How many decimal places should probability answers have?
The written probability quiz asks for numerical answers to at least four decimal places, and the solutions report probabilities that way throughout. Keep full precision during the calculation and round only the final answer.
Exam move
For every probability question, start by defining events with letters and translating the words into notation: union for “at least one”, intersection for “both”, a condition for “given” or “among”. Build small contingency tables of your own and compute every marginal, joint, union and conditional probability from them, then test independence both ways.
Write answers to four decimal places, as the written quiz requires. Keep one worked table of your own as a reference and recompute it from memory each week until no step needs checking. After each practice question, write a one-line justification for every probability: which rule, which denominator, which events.
Reviewing those lines shows whether errors come from reading the wording or from the arithmetic, and the fix is different for each.
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