LEARNING OBJECTIVES
By the end of this chapter, you should be able to:
- use the probability scale and complement rule
- estimate probability using relative frequency
- calculate expected frequency
- model combined events with diagrams
- distinguish mutually exclusive and independent events
- calculate conditional probabilities
THE BIG IDEA
Measure uncertainty, combine events and update probabilities when information changes.
Probability assigns a number from 0 to 1 to an uncertain event. Zero means impossible and one means certain.
A model is useful only when its outcomes and assumptions are clear. Tree, Venn and sample-space diagrams organise different types of information.
SECTION 01
Single events and complements
For equally likely outcomes, probability is favourable outcomes divided by total outcomes. Probabilities may be written as fractions, decimals or percentages.
The complement A′ contains every outcome in the universal set that is not in A.
Selecting a letter from MATHEMATICS
- There are 11 letters in total.
- The letter M appears twice.
- Assume each position is equally likely.
Answer: P(M) = 2/11 and P(not M) = 9/11.
SECTION 02
Relative and expected frequency
Relative frequency is experimental successes divided by trials. It estimates probability and usually becomes more stable as the number of trials increases.
Expected frequency predicts a long-run count by multiplying probability by the number of trials. It need not match one real experiment exactly.
A fair process gives intended outcomes equal chances; bias favours some outcomes; random means the next outcome cannot be known with certainty.
Estimating from an experiment
- A spinner lands on blue 87 times in 240 spins.
- Estimated P(blue) = 87/240 = 0.3625.
- For 800 future spins, expected blue results = 0.3625 × 800.
Answer: The expected frequency is 290.
SECTION 03
Combined events
A sample-space diagram lists ordered outcomes. A tree shows sequential events; probabilities along branches from one point sum to 1.
Multiply along a complete route and add probabilities of different routes that satisfy the event.
Without replacement, the total and relevant counts change after the first selection. With replacement, they reset.
KEY IDEAS
- For mutually exclusive A and B, P(A or B) = P(A) + P(B).
- For independent A and B, P(A and B) = P(A)P(B).
- In Venn diagrams, P(A ∪ B) means in at least one set and P(A ∩ B) means in both.
- For events that may overlap, P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Two counters without replacement
- A bag has 3 red and 2 blue counters.
- P(red then blue) = 3/5 × 2/4 = 3/10.
- P(blue then red) = 2/5 × 3/4 = 3/10.
- Add the two orders.
Answer: P(one of each colour) = 3/5.
SECTION 04
Conditional probability
Conditional probability restricts the sample space because some information is already known. Work only with outcomes inside the stated condition.
In a two-way table, use the conditioned row or column total as the new denominator. In a tree, follow the branch representing the known event.
Probability given a category
- A club has 18 juniors and 12 seniors.
- Seven juniors and nine seniors cycle to the club.
- A randomly chosen member is known to be a senior.
- Restrict the sample space to the 12 seniors.
Answer: P(cycles | senior) = 9/12 = 3/4.
QUICK CHAPTER SUMMARY
The ideas to carry forward
- Probability lies between 0 and 1.
- Complements sum to 1.
- Relative frequency estimates probability from data.
- Multiply along tree routes and add alternative successful routes.
- Replacement determines whether later probabilities change.
- Conditional information restricts the sample space.