POP

C8.1 Introduction to Probability

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01 — Explanation

What Does a Probability Between 0 and 1 Actually Mean?

Probability measures how likely an event is, on a scale from $0$ (impossible) to $1$ (certain):

   Impossible      Unlikely      Even chance      Likely        Certain
 ├─────────────┼─────────────┼─────────────┼─────────────┤
 0            0.25           0.5          0.75            1

Probabilities are always written as a fraction, a decimal, or a percentage — never as a ratio and never in words alone. An event with probability exactly $0.5$ is just as likely to happen as not; anything above $0.5$ is more likely to happen than not, and anything below is less likely.

02 — Explanation

How Do I Calculate the Probability of a Single Event?

When every outcome is equally likely:

P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}

A bag contains $8$ red, $5$ blue and $7$ yellow counters. Find $P(\text{blue})$ : total counters $=20$ , favourable outcomes $=5$ , so

P(\text{blue}) = \frac{5}{20} = \frac{1}{4}

Exam Tip

Always simplify the final fraction, and double-check the favourable outcomes are on top — a probability greater than $1$ is a sign the fraction has been written upside down.

03 — Explanation

How Do I Find a Probability From a List of Equally Likely Outcomes?

Write out (or count) every possible outcome first, then count how many of them satisfy the condition in the question.

A fair spinner is divided into $8$ equal sections, labelled $1$ to $8$ . Find $P(\text{score is a factor of }8)$ .

List the factors of $8$ from the possible outcomes $1$ – $8$ : these are $1, 2, 4, 8$ — four outcomes.
$P(\text{factor of }8) = \frac{4}{8} = \frac{1}{2}$

04 — Explanation

How Do I Use the Fact That "A" and "Not A" Together Cover Everything?

An event either happens or it doesn't, so the two probabilities must add to exactly $1$ :

P(\text{not } A) = 1 - P(A)

The probability that it rains tomorrow is $0.35$ . Find the probability it does not rain:

P(\text{no rain}) = 1 - 0.35 = 0.65

This rule is especially useful when the "not A" outcomes would be tedious to count directly — it's often far quicker to find $P(A)$ and subtract from $1$ than to list every outcome that doesn't satisfy $A$ .

05 — Explanation

How Do I Find a Probability From a Two-Way Table?

Read the relevant row/column total and the grand total directly from the table, then form the fraction.

	Walk	Bus	Total
Year 9	$18$	$12$	$30$
Year 10	$14$	$16$	$30$
Total	$32$	$28$	$60$

A student is picked at random from these $60$ . Find $P(\text{bus})$ :

P(\text{bus}) = \frac{28}{60} = \frac{7}{15}

Exam Tip

Read the total for the event itself (here, the "Bus" column total, $28$ ) rather than a single row's entry — a common mistake is to use just one year group's count (e.g. $12$ or $16$ ) instead of the combined column total.

06 — Explanation

Exam-Style Question

Using the same bag of $8$ red, $5$ blue and $7$ yellow counters, find the probability that a counter drawn at random is not red.

Solution

$P(\text{red}) = \dfrac{8}{20} = \dfrac{2}{5}$ .
Apply the complement rule:

P(\text{not red}) = 1-\frac{2}{5} = \frac{3}{5}

This can be checked directly: blue and yellow together make $5+7=12$ counters, and $\dfrac{12}{20}=\dfrac{3}{5}$ ✓ — matching answers confirm the complement rule was applied correctly.