MathsGCSE maths: ratio, probability and statistics30 min★★★ difficulty

Probability and interpreting statistics

Probability quantifies uncertainty and statistics summarises data. Both reward careful reading: the wording tells you which technique the question wants.

Lesson overview

What you'll learn in this lesson

Use proportional reasoning, probability and statistics to interpret and solve problems.

Key learning points

• Probability rules
• Tree and Venn diagrams
• Averages and spread

This lesson at a glance

30 minutes
16 parts to scroll through
4 quick checks
Marked quiz at the end
Gentle pace: short sittings with pauses

Words to know

proportionalreasoningprobabilitystatisticsinterpret

Scroll down — the lesson carries on below

Part 1 of 16 · Discover6%

Watch & discover

Part 1 of 16

Watch first

Probability and interpreting statistics

An English teaching video on this topic from mathantics. Watch it, then scroll down for the lesson.

In a nutshell

Use proportional reasoning, probability and statistics to interpret and solve problems.

Learning cycle

Part 2 of 16

Learning cycle 1 of 2

Part 1 · Probability rules

A short piece of teaching, then a check to make sure it has landed.

Explore the idea

Part 3 of 16

Learn

Probability rules

Probabilities of all outcomes sum to 1, so P(not A) = 1 − P(A). For independent events happening together, multiply. For mutually exclusive alternatives, add. 'At least one' is nearly always easiest as 1 minus the probability of none.

Explore the idea

Part 4 of 16

Learn

Tree and Venn diagrams

On a tree diagram, multiply along branches and add between them. Without replacement, the second set of denominators decreases by one, and this is where most marks are lost. A Venn diagram handles overlapping categories: fill the intersection first, then work outwards.

Quick check

Part 5 of 16

Quick check

P(not A) equals

Quick check

Part 6 of 16

Quick check

On a tree diagram you

Reset break

Part 7 of 16

Pause

That's sitting 1 of 2 done

Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.

Stop here for now

Learning cycle

Part 8 of 16

Learning cycle 2 of 2

Part 2 · Averages and spread

A short piece of teaching, then a check to make sure it has landed.

Explore the idea

Part 9 of 16

Learn

Averages and spread

The mean uses every value but is distorted by outliers; the median is resistant and better for skewed data; the mode suits categories. For grouped data, use midpoints to estimate the mean and say it is an estimate. Range and interquartile range describe spread, with the IQR ignoring extremes.

Quick check

Part 10 of 16

Quick check

Which average is least affected by an outlier?

Quick check

Part 11 of 16

Quick check

The mean from grouped data is an estimate because

Explore the idea

Part 12 of 16

Worked example

Worked answer: a bag holds 5 red and 3 blue counters. Two are taken without replacement. Find P(both red) (3 marks)

P(first red) = 5/8 (1). After removing a red counter, 4 reds remain out of 7 counters, so P(second red) = 4/7 (1). Multiplying along the branches: 5/8 × 4/7 = 20/56 = 5/14 (1). Using 5/8 twice would answer a different question, one with replacement.

Challenge round

Part 13 of 16

Game · Sort it

Which of these are true?

Drag each card into the right column. Tap a card first if dragging is fiddly.

True

Not true

Challenge round

Part 14 of 16

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

'At least one' is nearly always easiest as 1 minus the ____ of none.

Mastery quiz

Part 15 of 16

Marked quiz

End of lesson quiz: Probability and interpreting statistics

4 questions, marked with the reasoning shown. No timer.

1. P(not A) equals
2. On a tree diagram you
3. Which average is least affected by an outlier?
4. The mean from grouped data is an estimate because

Lesson round-up

Part 16 of 16

Lesson round-up

Ready when you are

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Do the work

Your written work for this lesson

This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.

Article

The cost of a school trip

A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.

If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.

The school decides to charge £20 and to use anything left over for a second trip later in the year.

Written for Good Learning Co.

Reading questions

Work out the total cost of the trip if 50 pupils go.
Show why the price per pupil falls when more pupils go, using numbers.
At £20 each, how many pupils must go before the trip makes money? Show your working.

Working out
Working out
Do the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.
10–15 questions
Worked example, explained
Writing
Copy one question out and write underneath, in words, why each step works.
1 question, 4–5 sentences
Problem in words
Working out
Write your own real-life problem that uses today's method, then solve it.
1 problem + solution

Write your answer, notes or working here

Saved as you type.

Photos of paper work, diagrams or working outHow many minutes did the written work take?

Part 1 of 16 · Watch & discover

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