Mr Kidman's Classroom
Revise the key ideas
Probability language and calculations
- A probability lies from 0 to 1 inclusive. Impossible events have probability 0; certain events have probability 1. A probability of 0.5 describes an even chance.
The probability scale - For equally likely outcomes, favourable outcomes are the ones matching the event you are counting. Then P(event) = . Count outcomes rather than assuming every named category is equally likely.
- Worked example: On a fair six-sided die, multiples of 3 are 3 and 6, so their probability is = . Fairness means each face has equal probability.
- The probabilities of an exhaustive set of mutually exclusive outcomes sum to 1. Exhaustive means all possibilities are covered; mutually exclusive means they cannot happen together.
- The complement of A is the event that A does not happen: P(not A) = 1 − P(A). If rain has probability 0.3, no rain has probability 0.7 under that model.
- For mutually exclusive events, P(A or B) = P(A) + P(B). If they overlap, simply adding double-counts the overlap; account for it using a Venn diagram.
Experiments and expected outcomes
- Relative frequency = . This estimates probability from observations; it is not necessarily the exact theoretical value.
- Worked example: A spinner lands on red 38 times in 100 spins. Estimated P(red) = 0.38. If this model holds, 250 future spins would have about 250 × 0.38 = 95 red results.
- Expected frequency = probability × number of trials. With P(success) = , 80 trials have expected frequency 20; the actual count can differ.
- More trials usually make an unbiased relative-frequency estimate more reliable. It tends towards the theoretical probability for a suitable stable random model, but need not get closer after every extra trial.
An unbiased estimation method gives estimates that are not systematically too high or too low over many repetitions.
- Small samples can give misleading proportions. Use a sufficiently large, fair experiment and keep the conditions consistent when estimating future outcomes.
- Independent trials, where one result does not change the next trial’s probabilities, do not “balance themselves out” on the next trial. After five heads on a fair coin, the next toss still has P(heads) = .
Organising evidence
- Frequency tables record counts by outcome. A frequency tree splits a total into groups; each parent's count must equal the sum of its branches.
Counts in a frequency tree - Worked example: Of 60 students, 35 walk and 25 use transport. If 20 walkers and 10 transport users bring lunch, the four leaf counts are 20, 15, 10 and 15. They total 60.
- State assumptions such as a fair die, a representative sample or unchanged conditions. A mathematically correct calculation can still be an unreliable prediction if its model is inappropriate.
