Probability, Scatter Diagrams and Exam Technique

This lesson covers DfE content statements L2.26, L2.27 and L2.28 — probability of single and combined events, two-way tables, tree diagrams, scatter diagrams and correlation — plus essential exam technique to help you perform at your best.

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Probability Basics

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

Probability	Meaning
0	Impossible
0.25 (1/4 or 25%)	Unlikely
0.5 (1/2 or 50%)	Even chance
0.75 (3/4 or 75%)	Likely
1	Certain

Probability = Number of favourable outcomes ÷ Total number of possible outcomes

Probability can be expressed as a fraction, decimal, or percentage.

Worked Example — Quality Control

Scenario: A factory produces 500 items. Quality testing shows 15 are defective. If one item is chosen at random, what is the probability it is defective?

P(defective) = 15/500 = 3/100 = 0.03 or 3%

Worked Example — Lottery-Style Draw

Scenario: A raffle has 200 tickets. You buy 8 tickets. What is the probability you win?

P(win) = 8/200 = 1/25 = 0.04 or 4%

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Complementary Events

If the probability of an event happening is P, then the probability of it NOT happening is 1 − P.

Worked Example

Scenario: The probability of rain tomorrow is 0.35. What is the probability it does NOT rain?

P(no rain) = 1 − 0.35 = 0.65

Exam Tip: Always check that your probabilities add up to 1 (or 100%). If they do not, you have made an error.

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Two-Way Tables

A two-way table organises data about two categories.

Worked Example — Staff Survey

Scenario: A company surveys employees about their preferred work pattern:

	Full-time	Part-time	Total
Office	45	20	65
Hybrid	35	25	60
Remote	15	10	25
Total	95	55	150

Questions:

(a) What is the probability that a randomly selected employee prefers hybrid working? P(hybrid) = 60/150 = 2/5 or 0.4

(b) Of those who work part-time, what fraction prefer remote working? P(remote | part-time) = 10/55 = 2/11

(c) What percentage of full-time workers prefer office working? 45/95 × 100 = 47.4%

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Combined Events — Sample Space Diagrams

When two events happen, list all possible outcomes in a sample space diagram.

Worked Example — Two Dice

Scenario: Two dice are thrown. Find the probability of getting a total of 7.

	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

Total outcomes = 36. Outcomes giving a total of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6

P(total 7) = 6/36 = 1/6

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Tree Diagrams

Tree diagrams show the probabilities of sequences of events. Multiply along branches, add between branches.

Worked Example — Weather Forecasting

Scenario: The probability of rain on Saturday is 0.4. If it rains Saturday, the probability of rain Sunday is 0.6. If it does not rain Saturday, the probability of rain Sunday is 0.2.

graph TD
    A[Start] -->|0.4 Rain| B[Saturday Rain]
    A -->|0.6 No rain| C[Saturday Dry]
    B -->|0.6 Rain| D["Rain both days<br/>0.4 × 0.6 = 0.24"]
    B -->|0.4 No rain| E["Sat rain, Sun dry<br/>0.4 × 0.4 = 0.16"]
    C -->|0.2 Rain| F["Sat dry, Sun rain<br/>0.6 × 0.2 = 0.12"]
    C -->|0.8 No rain| G["Dry both days<br/>0.6 × 0.8 = 0.48"]

Questions:

(a) P(rain both days) = 0.4 × 0.6 = 0.24

(b) P(rain on exactly one day) = 0.16 + 0.12 = 0.28

(c) P(no rain at all) = 0.6 × 0.8 = 0.48

Check: 0.24 + 0.16 + 0.12 + 0.48 = 1.00 ✓

Worked Example — Independent Events

Scenario: A bag contains 3 red and 5 blue marbles. A marble is drawn, replaced, then another is drawn. Find P(both red).

P(red) = 3/8 each time (replacement means probabilities stay the same). P(both red) = 3/8 × 3/8 = 9/64

Exam Tip: In tree diagrams, you MULTIPLY along branches (for "and") and ADD between final outcomes (for "or"). Always check your final probabilities sum to 1.

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Expected Frequency

Expected frequency = Probability × Number of trials

Worked Example

Scenario: A spinner has a 0.3 probability of landing on red. If spun 200 times, how many times would you expect it to land on red?

Expected = 0.3 × 200 = 60 times

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Scatter Diagrams and Correlation

A scatter diagram (scatter plot) shows the relationship between two variables by plotting data points on a graph.

Types of Correlation

Pattern	Name	Example
Points slope upward left to right	Positive correlation	Height and weight
Points slope downward left to right	Negative correlation	Car age and value
Points show no pattern	No correlation	Shoe size and IQ

Line of Best Fit

A line of best fit is a straight line drawn through the middle of the data points on a scatter diagram. It should have roughly equal numbers of points above and below.

Use the line of best fit to:

• Interpolate (estimate within the data range — reliable)
• Extrapolate (estimate beyond the data range — less reliable)

Worked Example — Ice Cream Sales

Scenario: Data shows the temperature and number of ice creams sold at a kiosk: