Probability basics | AQA GCSE Maths Notes

Visual model

Probability sits between 0 and 1

0

impossible

\frac{1}{2}

even chance

1

certain

Gold-standard guide

20 mins

What you will learn

Describe chance on a scale from impossible to certain.

Use a clear step-by-step method for probability basics.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier probability skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Probability = number of favourable outcomes / total number of outcomes

Step 1

Find the total number of outcomes

5 + 3 + 2 = 10 marbles

Step 2

Find P(red)

P(red) = $\frac{5}{1}0$ = $\frac{1}{2}$

Step 3

Find P(not red) using the complement

P(not red) = 1 − $\frac{1}{2}$ = $\frac{1}{2}$

Watch out

Students add probabilities for events that are not mutually exclusive (for example P(even) + P(greater than 3) on a die — these overlap)

Probability

$probability = favourable outcome\frac{s}{total} outcomes.$

Complement

$P(not A) = 1 - P(A).$

Worked example

A bag contains 5 red, 3 blue and 2 green marbles. A marble is chosen at random. Find P(red), P(not red) and P(blue or green).

Find the total number of outcomes: 5 + 3 + 2 = 10 marbles.

Find P(red): P(red) = $\frac{5}{1}0$ = $\frac{1}{2}$ .

Find P(not red) using the complement: P(not red) = 1 − $\frac{1}{2}$ = $\frac{1}{2}$ . Or directly: 5 non-red out of 10 = $\frac{5}{1}0$ = $\frac{1}{2}$ .

Find P(blue or green): Blue and green are mutually exclusive. P(blue or green) = $\frac{3}{1}0$ + $\frac{2}{1}0$ = $\frac{5}{1}0$ = $\frac{1}{2}$ .

Final answer

P(red) = $\frac{1}{2}$ ; P(not red) = $\frac{1}{2}$ ; P(blue or green) = $\frac{1}{2}$

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

A bag contains 5 red, 3 blue and 2 green marbles. A marble is chosen at random. Find P(red), P(not red) and P(blue or green).

4 marks4 minsprobability-basics-worked

Show solution

Worked solution

1.Find the total number of outcomes: 5 + 3 + 2 = 10 marbles.
2.Find P(red): P(red) = $\frac{5}{1}0$ = $\frac{1}{2}$ .
3.Find P(not red) using the complement: P(not red) = 1 − $\frac{1}{2}$ = $\frac{1}{2}$ . Or directly: 5 non-red out of 10 = $\frac{5}{1}0$ = $\frac{1}{2}$ .
4.Find P(blue or green): Blue and green are mutually exclusive. P(blue or green) = $\frac{3}{1}0$ + $\frac{2}{1}0$ = $\frac{5}{1}0$ = $\frac{1}{2}$ .

Final answer

P(red) = $\frac{1}{2}$ ; P(not red) = $\frac{1}{2}$ ; P(blue or green) = $\frac{1}{2}$

Mark points

M1: find the total number of outcomes
M1: find p(red)
M1: find p(not red) using the complement
M1: find p(blue or green)
A1: P(red) = $\frac{1}{2}$ ; P(not red) = $\frac{1}{2}$ ; P(blue or green) = $\frac{1}{2}$

Watch out

g. P(even) + P(greater than 3) on a die — these overlap). Only add if events cannot both happen.Use a Venn diagram or list outcomes to check for overlaps.

Diagnosticrecall

A fair six-sided die is rolled. Find P(prime).

1 mark2 minsprobability-basics-q1

Show solution

Worked solution

1.Spot the skill: Probability = number of favourable outcomes / total number of outcomes.
2.Use the find the total number of outcomes stage first, then find p(red).
3.Keep the final answer visible: $\frac{3}{6}$ = $\frac{1}{2}$ .

Final answer

$\frac{3}{6}$ = $\frac{1}{2}$

Mark points

M1: use the correct probability = number of favourable outcomes / total number of outcomes.probabilities range from 0 (impossible) to 1 (certain). p(not a) = 1 − p(a).add probabilities for mutually exclusive events (cannot both happen). never add for events that can overlap.
A1: $\frac{3}{6}$ = $\frac{1}{2}$

Watch out

g. P(even) + P(greater than 3) on a die — these overlap). Only add if events cannot both happen.Use a Venn diagram or list outcomes to check for overlaps.

Easyprocedure

P(A) = 0.35. Find P(not A).

2 marks3 minsprobability-basics-q2

Show solution

Worked solution

1.Spot the skill: Probability = number of favourable outcomes / total number of outcomes.
2.Use the find p(red) stage first, then find p(not red) using the complement.
3.Keep the final answer visible: 0.65.

Final answer

0.65

Mark points

M1: use the correct probability = number of favourable outcomes / total number of outcomes.probabilities range from 0 (impossible) to 1 (certain). p(not a) = 1 − p(a).add probabilities for mutually exclusive events (cannot both happen). never add for events that can overlap.
A1: 0.65

Watch out

g. P(even) + P(greater than 3) on a die — these overlap). Only add if events cannot both happen.Use a Venn diagram or list outcomes to check for overlaps.

Mediumreasoning

Cards 1 to 20: find P(multiple of 4 or multiple of 6).

3 marks4 minsprobability-basics-q3

Show solution

Worked solution

1.Spot the skill: Probability = number of favourable outcomes / total number of outcomes.
2.Use the find p(not red) using the complement stage first, then find p(blue or green).
3.Keep the final answer visible: $\frac{7}{2}0$ (multiples of 4: 4,8,12,16,20; of 6: 6,12,18; overlap at 12 — 7 distinct values).

Final answer

$\frac{7}{2}0$ (multiples of 4: 4,8,12,16,20; of 6: 6,12,18; overlap at 12 — 7 distinct values)

Mark points

M1: use the correct probability = number of favourable outcomes / total number of outcomes.probabilities range from 0 (impossible) to 1 (certain). p(not a) = 1 − p(a).add probabilities for mutually exclusive events (cannot both happen). never add for events that can overlap.
A1: $\frac{7}{2}0$ (multiples of 4: 4,8,12,16,20; of 6: 6,12,18; overlap at 12 — 7 distinct values)

Watch out

g. P(even) + P(greater than 3) on a die — these overlap). Only add if events cannot both happen.Use a Venn diagram or list outcomes to check for overlaps.

Hardproblem solving

A spinner has P(red) = 0.3, P(blue) = 0.45. Find P(green) if those are the only colours.

3 marks5 minsprobability-basics-q4

Show solution

Worked solution

1.Spot the skill: Probability = number of favourable outcomes / total number of outcomes.
2.Use the find p(blue or green) stage first, then find the total number of outcomes.
3.Keep the final answer visible: 0.25.

Final answer

0.25

Mark points

M1: use the correct probability = number of favourable outcomes / total number of outcomes.probabilities range from 0 (impossible) to 1 (certain). p(not a) = 1 − p(a).add probabilities for mutually exclusive events (cannot both happen). never add for events that can overlap.
A1: 0.25

Watch out

g. P(even) + P(greater than 3) on a die — these overlap). Only add if events cannot both happen.Use a Venn diagram or list outcomes to check for overlaps.

Exam-stylemulti-step

A bag has r red and 3 blue balls. P(red) = $\frac{2}{5}$ . Find r.

4 marks6 minsprobability-basics-q5

Show solution

Worked solution

1.Spot the skill: Probability = number of favourable outcomes / total number of outcomes.
2.Use the find the total number of outcomes stage first, then find p(red).
3.Keep the final answer visible: r = 2.

Final answer

r = 2

Mark points

M1: use the correct probability = number of favourable outcomes / total number of outcomes.probabilities range from 0 (impossible) to 1 (certain). p(not a) = 1 − p(a).add probabilities for mutually exclusive events (cannot both happen). never add for events that can overlap.
A1: r = 2

Watch out

g. P(even) + P(greater than 3) on a die — these overlap). Only add if events cannot both happen.Use a Venn diagram or list outcomes to check for overlaps.

Grade 9 stretchproblem solving

Two fair six-sided dice are rolled. Find the probability that the total is 10.

4 marks7 minsprobability-g9

Show solution

Worked solution

1.List the successful ordered pairs.
2.Divide by the 36 equally likely outcomes.

Final answer

$\frac{1}{1}2$

Mark points

M1: identify (4,6), (5,5), (6,4)
A1: $\frac{3}{3}6$ = $\frac{1}{1}2$

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Probability basics - 2 marksA fair six-sided die is rolled. Find P(prime).Mark answer

Answer

$\frac{3}{6}$ = $\frac{1}{2}$

2Relative frequency - 2 marksA spinner is spun 500 times. Sector A comes up 175 times. Estimate P(A).Mark answer

Answer

0.35

3Sample spaces and frequency trees - 2 marksA bag has 4 red, 2 blue balls. Two are drawn with replacement. Find P(both red).Mark answer

Answer

$\frac{4}{6}$ × $\frac{4}{6}$ = $\frac{4}{9}$

4Venn diagrams and set notation - 3 marksIn a Venn diagram, the 'only A' region has 9, 'only B' has 7, 'both' has 4. Find P(A).Mark answer

Answer

P(A) = $1\frac{3}{2}0$

Mastery check

I can explain the method for probability basics.
I can show clear working without skipping key steps.
g. P(even) + P(greater than 3) on a die — these overlap). Only add if events cannot both happen.Use a Venn diagram or list outcomes to check for overlaps.

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