E8.1 Introduction to probability

Learning objectives

E8.1.1Understand and use the probability scale from 0 to 1.• Understand and use the probability scale from 0 to 1.E8.1.2Understand and use probability notation.• Understand and use probability notation.E8.1.3Calculate the probability of a single event.• Calculate the probability of a single event.E8.1.4Understand that the probability of an event not occurring = 1 – the• Understand that the probability of an event not occurring = 1 – the probability of the event occurring. P(A) is the probability of A P(A′) is the probability of not A Probabilities should be given as a fraction, decimal or percentage. Problems may require using information from tables, graphs or Venn diagrams. e.g. P(B) = 0.8, find P(B′)

Probability measures likelihood on a scale from 0 to 1 inclusive. A larger value means the event is more likely.

Probability	Meaning
$0$	impossible
between $0$ and $0.5$	unlikely
$0.5$	equally likely to occur or not occur
between $0.5$ and $1$	likely
$1$	certain

Convert values to the scale's form before locating or comparing them. For example, $\tfrac38=0.375$ lies between 0 and 0.5, while $65\%=0.65$ lies between 0.5 and 1. The larger probability is farther to the right.

The endpoints 0 and 1 are valid probabilities. No probability can be negative or exceed 1, and a probability describes likelihood rather than guaranteeing how often an event occurs in a short experiment.

An event is a set of outcomes and is usually named with a capital letter. $P(A)$ means the probability that event $A$ occurs.

Notation	Meaning
$A$	the event itself
$P(A)$	the probability that $A$ occurs
$A'$	the complementary event: $A$ does not occur
$P(A')$	the probability that $A$ does not occur

Let $A$ be the event 'a fair six-sided die shows an even number'. The favourable outcomes are 2, 4 and 6, so $P(A)=\tfrac36=\tfrac12$ . Event $A'$ is 'the die shows an odd number', and $P(A')=\tfrac12$ .

$P(A)$ is a number between 0 and 1; it is not $P$ multiplied by $A$ . Define the event before using its letter, and do not confuse $A'$ with a second unrelated event.

When individual outcomes are equally likely, divide the number of outcomes in the event by the total number of possible outcomes.

P(A)=\frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}

Identify the complete outcome set, count each outcome once, select only those satisfying the event, then simplify the fraction or convert it to a decimal or percentage if required. Counts may come from a table, graph or Venn diagram.

A bag contains 4 black, 3 white and 5 yellow counters. If one counter is selected at random, there are 12 equally likely counters and 8 are not yellow. Therefore $P(\text{not yellow})=\tfrac8{12}=\tfrac23$ .

The favourable-over-total rule requires equally likely individual outcomes. Do not use the number of category labels as the denominator when categories contain different numbers of outcomes.

Event $A$ and its complement $A'$ cover every possible outcome exactly once, so their probabilities add to 1.

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

Given	Complement
$P(A)=\tfrac7{10}$	$P(A')=\tfrac3{10}$
$P(B)=0.8$	$P(B')=0.2$
$P(C)=18\%$	$P(C')=82\%$

With counts, subtract the event count from the total before dividing. In a table, graph or Venn diagram, include every region outside the event; those regions together form the complement.

$A'$ means every allowed outcome not in $A$ , not one chosen alternative. Subtract from 1 for fractions or decimals and from $100\%$ for percentages; the two probabilities must sum to one whole.

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—E8.2 Relative and expected frequenciesRead

E8.1 Introduction to probability

Learning objectives

Read and use the probability scale

Use probability notation for events

Calculate the probability of one event

Find the probability of a complementary event

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