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Chance: cheat sheet

Data and chance · The words and methods from the lessons, with an example of each.

sample space

Every possible outcome, listed once. Check whether the outcomes are equally likely.

{1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\}

expected count

Expected count = probability × trials. Real results vary around it, more so in short experiments.

60×16=1060 \times \tfrac{1}{6} = 10

relative frequency

The experiment's estimate of the probability. Over many trials it settles close to the probability (the law of large numbers); nothing is ever 'due'.

counttrials=1320=0.65\dfrac{\text{count}}{\text{trials}} = \dfrac{13}{20} = 0.65
2000 flips: the relative frequency of heads settles near 0.5
2000 flips: the relative frequency of heads settles near 0.5

Venn diagram and two-way table

Two yes/no properties make four groups: one only, both, the other only, neither. Each is a region of the Venn diagram and a cell of the table.

both first: only=total−both\text{both first: only} = \text{total} - \text{both}
The same 30 students as a Venn diagram and a two-way table
The same 30 students as a Venn diagram and a two-way table

and, or, not

And is the overlap; or is one, the other or both; not A is everything outside A. Count each outcome once.

P(A or B)=P(A)+P(B)−P(A and B)P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
Dog or cat: everything inside the circles, 19 of 30
Dog or cat: everything inside the circles, 19 of 30

probability

For equally likely outcomes. Otherwise use sizes or counts. From 0 (impossible) to 1 (certain); all outcomes add to 1.

P(even)=favourabletotal=36=12P(\text{even}) = \dfrac{\text{favourable}}{\text{total}} = \dfrac{3}{6} = \dfrac{1}{2}
A half-red spinner: P(red) = 1/2, P(blue) = P(green) = 1/4
A half-red spinner: P(red) = 1/2, P(blue) = P(green) = 1/4

complementary events

An event and its complement cover every outcome once. For 'at least one', count the outcomes with none and subtract from 1.

P(not A)=1−P(A)P(\text{not } A) = 1 - P(A)
Roll 5 or 6 (A) and the other four outcomes (not A)
Roll 5 or 6 (A) and the other four outcomes (not A)

grid and tree diagram

Two events make pairs: a grid or a tree lists every pair once, and the number of outcomes is the two counts multiplied. HT and TH are different outcomes.

outcomes=3×2=6\text{outcomes} = 3 \times 2 = 6
A spinner with three equal parts, then a coin: 6 outcomes, 3 of them heads
A spinner with three equal parts, then a coin: 6 outcomes, 3 of them heads

simulation

Run many trials with random numbers (=RANDBETWEEN(1,6) is a dice) and compare the relative frequency with the probability. Short runs can be well off.

relative frequency=counttrials\text{relative frequency} = \dfrac{\text{count}}{\text{trials}}
Two dice with a total of 10 or more chosen (6 of 36), ready to simulate
Two dice with a total of 10 or more chosen (6 of 36), ready to simulate