Words to know
equally likely outcomes
Outcomes that all have the same chance of happening, such as the faces of a fair die.
theoretical probability
For equally likely outcomes, P(A) = n(A)/n(S): the number of outcomes in event A divided by the number of outcomes in the sample space.
fair
Describes a die, coin, spinner or deck in which every outcome is equally likely.
probability distribution
A table, graph or formula that gives the probability of every outcome (or every value) of an experiment. The probabilities add to 1.
random
Chosen so that every outcome has the probability given by the model, with no pattern or influence.
The counting formula
When every outcome in a sample space is equally likely, the theoretical probability of an event A is
P(A) = n(A) / n(S)
where n(A) is the number of outcomes in A (the "favourable" outcomes) and n(S) is the total number of outcomes.
For a fair die, P(even) = n({2, 4, 6}) / n({1, 2, 3, 4, 5, 6}) = 3/6 = 1/2.
The formula only works when the outcomes really are equally likely. That is why we use fair dice, well-shuffled decks and random selection. If a die is weighted, or a spinner has sections of different sizes, you need a different model.
Probabilities add to 1
List every outcome of a sample space and its probability. Since something must happen, and outcomes do not overlap, the probabilities must add to exactly 1:
P(outcome 1) + P(outcome 2) + … + P(last outcome) = 1
A list or table of all outcomes with their probabilities is called a probability distribution. Checking that it adds to 1 is the best way to catch a counting mistake. (In the unit Discrete Probability Distributions you will give these a full treatment, with expected values and the binomial and hypergeometric distributions.)
When outcomes are not equally likely
A spinner with sectors of different sizes does not have equally likely outcomes, but you can still find theoretical probabilities by reasoning: if the pointer is equally likely to stop at any angle, then P(sector) = sector angle ÷ 360°. The probability distribution must still add to 1.
Hands-on activityThe difference of two dice
You need: two dice (or two paper cubes numbered 1 to 6), paper and a pencil.
- Draw a 6 × 6 table like the one in lesson 2, but in each cell write the difference larger − smaller (for example (2, 5) gives 3 and (4, 4) gives 0).
- Count how many cells give each difference 0, 1, 2, 3, 4, 5. Write the probability distribution and check that it adds to 1.
- Which difference is most likely? Make a prediction for 36 rolls.
- Roll the two dice 36 times and tally the differences. Compare with your prediction.
What to look for: difference 0: 6 ways, 1: 10, 2: 8, 3: 6, 4: 4, 5: 2. Total 36, so the probabilities are 6/36, 10/36, 8/36, 6/36, 4/36 and 2/36 and add to 1. A difference of 1 is most likely (10/36 ≈ 0.28). In 36 rolls you expect about 10 differences of 1, but your count will vary.