P1, P2, P3, P4ProbabilityFoundationGrades 1–4

Basic Probability

Probability measures how likely something is, on a scale from 0 (impossible) to 1 (certain). When outcomes are equally likely, a probability is simply a fraction: the number of ways the event can happen over the total number of outcomes. This lesson builds the foundations every later probability topic depends on.

35 min AQA GCSE Maths

The big picture

Probabilities are always between 0 and 1 and can be written as fractions, decimals or percentages — never as "1 in 4" or "3 : 1" in an exam answer. The probabilities of all the possible outcomes of an experiment add up to exactly 1.

That sum-to-1 fact is the most useful tool in the topic. It gives $P(\text{not } A) = 1 - P(A)$ , and it lets you find a missing probability in a table — often with algebra.

What you'll learn

Use the probability scale and the language of likelihood
Calculate probabilities of equally likely outcomes
Use $P(\text{not } A) = 1 - P(A)$ and that probabilities sum to 1
Calculate expected outcomes

The probability scale

An event with probability 0 is impossible; 1 is certain; $\frac{1}{2}$ is an even chance. Words such as "unlikely" and "likely" describe positions on the scale.

For equally likely outcomes: $P(\text{event}) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$ .

Rolling anything other than a 6 on a fair dice is likely:

\frac{5}{6}

Example 1A bag has 4 red, 7 blue and 9 green counters. One is chosen at random. Find

P(\text{blue})

1Total counters:

4 + 7 + 9 = 20

27 of them are blue.

Answer

\frac{7}{20}

Your turnSame method, new numbers — try it before moving on.

A letter is chosen at random from the word PROBABILITY. What is the probability it is a B?

Your answer

Show worked solution

Solution

1PROBABILITY has 11 letters.
2It contains 2 Bs: $P(B) = \frac{2}{11}$ .

Answer: $\frac{2}{11}$

Not happening and summing to 1

$P(\text{not } A) = 1 - P(A)$ , because an event either happens or it does not.

If all the outcomes of an experiment are listed in a table, their probabilities add up to 1. Use that to find a missing value — including when the probabilities are given in terms of $x$ .

Example 1The probability that a train is late is 0.15. What is the probability that it is not late?

1 - 0.15 = 0.85

Answer

0.85

Your turnSame method, new numbers — try it before moving on.

The probability of rain tomorrow is 32%. What is the probability, as a decimal, that it does not rain?

Your answer

Show worked solution

Solution

1 $1 - 0.32 = 0.68$ .

Answer: $0.68$

Example 2A spinner lands on red, blue, green or yellow with probabilities

0.3

0.25

x

and

2x

. Find

x

0.3 + 0.25 + x + 2x = 1

3x = 0.45

, so

x = 0.15

Answer

x = 0.15

Your turnSame method, new numbers — try it before moving on.

Outcomes A, B, C and D have probabilities $0.4$ , $x$ , $x$ and $0.2$ . Find $x$ .

Your answer

Show worked solution

Solution

1 $0.4 + 2x + 0.2 = 1$ .
2 $2x = 0.4$ , so $x = 0.2$ .

Answer: $x = 0.2$

Tip — Check that your final probabilities all lie between 0 and 1 — a negative or bigger-than-1 value signals an error.

Expected outcomes

Expected number of times an event happens $=$ probability $\times$ number of trials. It is what you would expect on average, not a guarantee.

Example 1A fair six-sided dice is rolled 300 times. How many sixes would you expect?

P(6) = \frac{1}{6}

\frac{1}{6} \times 300 = 50

Answer

50

Your turnSame method, new numbers — try it before moving on.

The probability a seed germinates is 0.85. A gardener plants 240 seeds. How many would she expect to germinate?

Your answer

Show worked solution

Solution

1 $0.85 \times 240 = 204$ .

Answer: $204$

Expected is not guaranteed

Rolling a dice 6 times will often not give exactly one 6. Expected outcomes describe the long-run average — the more trials, the closer the results tend to get. That idea leads directly to relative frequency.

Think like an examiner

Write probabilities as fractions, decimals or percentages. "3 out of 10" or "3 : 10" may not be accepted.
Fractions do not need simplifying unless asked, but they must be correct.
In "find $x$ " questions, show the equation that sums to 1.

Probability rules

P(A) = \dfrac{\text{favourable}}{\text{total}}

Equally likely outcomes.

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

Complement.

\text{expected} = P \times n

Expected outcomes.

Watch out for these

Writing a probability as "1 in 6".

Write

\frac{1}{6}

P(\text{not blue}) = \frac{1}{7}

when

P(\text{blue}) = \frac{7}{20}

1 - \frac{7}{20} = \frac{13}{20}

Using the number of colours as the total outcomes.

Use the total number of individual items.

Expected 50 sixes means exactly 50 will happen.

It is an average expectation, not a certainty.

Stretch yourself

A bag contains only red and blue beads. The probability of picking red is $\frac{3}{8}$ . When 10 more red beads are added, the probability of picking red becomes $\frac{1}{2}$ . How many beads were in the bag at the start?

Hint — Let there be $8k$ beads: $3k$ red and $5k$ blue. Form an equation after adding 10 red.

Reveal solution

SolutionRed

= 3k

, blue

= 5k

. After adding:

\frac{3k + 10}{8k + 10} = \frac{1}{2}

, so

6k + 20 = 8k + 10

, giving

k = 5

. There were

8 \times 5 = 40

beads. Check: 25 red out of 50

= \frac{1}{2}

✓.

\blacksquare

Questions students ask

What does "at random" mean?

Every item has the same chance of being chosen, so outcomes are equally likely.

Can I give a probability as a percentage?

Yes — fractions, decimals and percentages are all acceptable. Just be consistent within a question.

What is the difference between theoretical and experimental probability?

Theoretical probability comes from equally likely outcomes (like a fair dice). Experimental probability, or relative frequency, comes from results of trials.

Key takeaways

Probabilities run from 0 (impossible) to 1 (certain).
Equally likely: favourable outcomes over total outcomes.
$P(\text{not } A) = 1 - P(A)$ .
All outcome probabilities sum to 1.
Expected outcomes $=$ probability $\times$ trials.

Exam-style questions

Practise it the way it is examined

2 original questions · 6 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 22 marks

A box contains 5 milk, 8 dark and 3 white chocolates. Jess takes one at random.

(a)Write down the probability that it is dark.[1]
Show mark scheme (a)
- B1 $\frac{8}{16}$ or $\frac{1}{2}$ or equivalent
Answer: $\frac{1}{2}$
(b)Write down the probability that it is not white.[1]
Show mark scheme (b)
- B1 $\frac{13}{16}$ or equivalent
Answer: $\frac{13}{16}$

Question 2Grade 44 marks

A biased spinner has four sections. The table shows some probabilities. $\quad \begin{array}{c|c|c|c} 1 & 2 & 3 & 4 \\ \hline 0.35 & 0.25 & x & 3x \end{array}$

(a)Work out the value of $x$ .[2]
Show mark scheme (a)
- M1 $0.35 + 0.25 + 4x = 1$
- A1 $x = 0.1$
Answer: $x = 0.1$
(b)The spinner is spun 400 times. How many times would you expect it to land on 4?[2]
Show mark scheme (b)
- M1 $0.3 \times 400$
- A1 $120$
Answer: $120$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

The probability of an impossible event is…
- $0$
- $1$
- $\frac{1}{2}$
- $-1$
Show answer and explanation
Answer: $0$
Impossible is 0 on the scale.
P(even number) on a fair six-sided dice?
- $\frac{1}{2}$
- $\frac{1}{3}$
- $\frac{1}{6}$
- $\frac{2}{3}$
Show answer and explanation
Answer: $\frac{1}{2}$
2, 4, 6 out of 6.
$P(A) = 0.7$ . What is $P(\text{not } A)$ ?
- $0.3$
- $0.7$
- $1.7$
- $0.07$
Show answer and explanation
Answer: $0.3$
$1 - 0.7$ .
Which could NOT be a probability?
- $1.2$
- $0.99$
- $\frac{3}{7}$
- $45\%$
Show answer and explanation
Answer: $1.2$
Probabilities cannot exceed 1.
A card is drawn from a standard 52-card pack. P(a king)?
- $\frac{1}{13}$
- $\frac{1}{4}$
- $\frac{1}{52}$
- $\frac{4}{13}$
Show answer and explanation
Answer: $\frac{1}{13}$
4 kings out of 52.
P(heads) $= 0.5$ . Expected heads in 90 flips?
- $45$
- $50$
- $90$
- $0.5$
Show answer and explanation
Answer: $45$
$0.5 \times 90$ .
Probabilities $0.2$ , $x$ , $0.5$ sum to 1. $x =$
- $0.3$
- $0.7$
- $0.2$
- $0.5$
Show answer and explanation
Answer: $0.3$
$1 - 0.7$ .
P(red) $= \frac{2}{5}$ and there are 12 red balls. How many balls in total?
- $30$
- $24$
- $60$
- $20$
Show answer and explanation
Answer: $30$
$\frac{2}{5}$ of 30 is 12.

Next step

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