Calculating probabilities and sample space

Probability as the proportion of equally likely outcomes that are favourable; listing the sample space (including for two dice and two-way tables); the fact that probabilities of all outcomes sum to 1, and the complement rule P(A′) = 1 − P(A); and the difference between theoretical probability and experimental (relative frequency) probability.

Statistics · Probability

Calculating probabilities and sample space

Why it works

A probability measures how likely an event is, on a scale from

0

(impossible) to

1

(certain). When every outcome of an experiment is equally likely,

P(\text{event}) = \frac{\text{number of outcomes in the event}}{\text{total number of possible outcomes}}.

The list of all possible outcomes is the sample space. Getting the probability right almost always comes down to listing — or counting — the sample space correctly.

A single experiment. Rolling a fair die has sample space

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

, six equally likely outcomes. The event "even"

= \{2,4,6\}

, so

P(\text{even}) = \tfrac{3}{6} = \tfrac{1}{2}

.

Two experiments at once. For two fair dice, the sample space is every ordered pair

(a,b)

— a

6\times 6

grid of

36

equally likely outcomes. To find

P(\text{sum} = 7)

, count the pairs that total

7

(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)

— six of them — so

P = \tfrac{6}{36} = \tfrac{1}{6}

. The grid is the safe way to avoid miscounting.

The total is always 1. The probabilities of all the outcomes in a sample space add up to

1

. This gives the complement rule: the event "not

A

", written

A'

, has

P(A') = 1 - P(A).

This is often the quickest route: "the probability of at least one" is usually found as

1 - P(\text{none})

.

Two-way tables. When data is cross-classified (e.g. by gender and by choice), a two-way table is the sample space — each person sits in exactly one cell. A probability is then (the relevant total) ÷ (the overall total). For

80

people with

30

in a cell, the probability of landing in that cell is

\tfrac{30}{80} = 0.375

.

Theoretical vs experimental. A theoretical probability comes from assuming outcomes are equally likely (a fair die gives

\tfrac16

). An experimental probability (or relative frequency) comes from actually doing the experiment:

\dfrac{\text{number of times it happened}}{\text{number of trials}}

. For a fair object the relative frequency settles near the theoretical value as the number of trials grows; a big mismatch is evidence the object is biased.