S3 Maths | Introduction to Probability Lesson-Prep Guide (for tutors)

S3 Maths | Introduction to Probability Lesson-Prep Guide (for tutors) | Sample Space, Complement

2026-11-10

Tutor lesson-prep guides · Secondary Maths (DSE)

The denominator is not how many outcomes you can name, but how many equally likely outcomes exist; that single sentence decides every question.

Objectives — Students can (1) list the sample space for one or two dice and explain why each outcome is equally likely; (2) compute probability as favourable outcomes over all equally likely outcomes and add correctly for mutually exclusive events; (3) use the complement rule and expected frequency in context, scoring at least 9 of 12 diagnostic questions.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Simplifying fractions	Simplify 3 over 6.	1 over 2
Listing	How many outcomes are there for two dice?	36
Percentages	Write 0.4 as a percentage.	40 percent

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-6 min	Write the full sample space	One die gives 6 outcomes; two dice give 36 ordered outcomes, not 11 sums	Build the table
6-16 min	Classical probability	An even number has 3 favourable outcomes, so the probability is 0.5	Write numerator and denominator
16-28 min	Mutually exclusive events	Sum 7 has 6 outcomes and sum 12 has 1; they cannot overlap, so add	Check for overlap first
28-40 min	Complement rule	Rain has probability 0.35, so no rain is 0.65; about 7 days in 20	Use one minus P(A)
40-54 min	12-question diagnostic	Three coins, a spinner and a bag of balls: 9 correct of 12	Work independently
54-60 min	Exit check	Three oral questions; any wrong answer sends the student back	Answer orally

03 Teaching script (read it out)

Scene 1: A sample space must be complete and equally likely

For two dice, listing “sums from 2 to 12, eleven outcomes” is wrong: sum 7 has 6 ways and sum 12 has only 1, so the sums are not equally likely. The correct space has 36 ordered outcomes, each with probability 1 over 36.
Teaching move: have students draw a six by six grid, first die down the side and second along the top, and count every later question on that grid.

Students do: fill a 36-cell grid for two dice and count cells with sum 7 (6) and sum 12 (1).

Scene 2: Classical probability counts outcomes, not objects

An even number on one die has 2, 4 and 6, that is 3 favourable outcomes out of 6, giving one half. A probability always lies between 0 and 1, so an answer above 1 or a denominator smaller than the numerator signals a counting error.
Teaching move: write the denominator first, then the numerator, then simplify; a fixed order removes most careless mistakes.

Students do: find the probability of a prime on one die, writing the full outcome list and the favourable count.

Scene 3: Only mutually exclusive events may be added

Sum 7 and sum 12 cannot happen together, so their probabilities add: 36ths gives 7 over 36. Events such as “even” and “greater than 3” do overlap at 4 and 6, so adding them directly counts those outcomes twice.
Teaching move: before adding, ask whether any single outcome belongs to both events, and subtract the overlap if it does.

Students do: find the probability that two dice total 10 or more (6 over 36, which is one sixth) and justify the addition.

Scene 4: Complement and expected frequency

When counting the opposite event is easier, use one minus the probability: rain is 0.35, so no rain is 0.65. Expected frequency multiplies trials by probability: a spinner wins 40 percent of the time, so 300 spins give about 120 wins.
Teaching move: stress that expected means long-run average, not a guarantee, and that 300 spins will not give exactly 120.

Students do: for three coins, list 8 outcomes, find exactly two heads (3 over 8) and check with the complement of zero or three heads.

04 Question types and worked steps

Type	Example	Steps	Answer
Single event	even number on one die	favourable over all equally likely	one half
Two dice	sum 7 and sum 12	use 36 ordered outcomes, never 11 sums	6 over 36 and 1 over 36
Mutually exclusive	sum 7 or sum 12	confirm no overlap, then add	7 over 36
Expected frequency	40 percent spinner, 300 spins	trials times probability, answer says about	about 120

Sanity check (teach this): Sanity checks: every probability lies between 0 and 1; a complete set of mutually exclusive outcomes sums to 1; using 36 ordered outcomes means the denominator is 36, never 12 or 21; an expected frequency cannot exceed the number of trials; the word “about” belongs in every expected-value answer; 9 of 12 is 75 percent, and below that the sample space must be retaught.

05 Independent practice (3 questions)

Question	Answer	What to watch
Four single events	dice, playing cards, a ball from a bag	is the denominator all equally likely outcomes
Three two-dice questions	a given sum, a given product	did they count on the 36-cell grid
Three complement questions	at least one, not a given outcome	did they use one minus P(A) instead of listing
Two expected-frequency questions	spinner and raffle	does the answer say about

06 Common mistakes and what to say

Mistake	Student writes	Say this
Using sums as outcomes	eleven outcomes for two dice	Sum 7 has six ways and sum 2 has one, so they are not equally likely; use all 36 ordered outcomes.
Putting the total in the denominator	an even number written as one third	The denominator is all 6 outcomes, not 3; 3 is the number of favourable outcomes.
Adding overlapping events	adding “even” and “greater than 3”	The outcomes 4 and 6 belong to both events, so subtract them before you add.
Treating expected as guaranteed	claiming exactly 120 wins from 300 spins	Expected frequency is a long-run average; with few trials the result can be far off, so write “about”.

07 Homework (paper, section and question numbers)

Secondary maths index — See the other chapters in this subject
All lesson-prep guides — Compare related chapters
Redo this chapter — Redo diagnostic questions 7 to 12, writing the sample space first

08 Exit check (last 5 minutes)

Question	Answer	Decision
Probability of an even number on one die	one half	Answering one third means the denominator is wrong, retake part two
Probability of sum 7 with two dice	6 over 36	Answering one over 11 means 11 sums were used, retake part one
Probability of no rain when rain is 0.35	0.65	Answering 0.35 means the complement was not used, retake part four

09 Tutor FAQ

Why are there 36 outcomes for two dice?

The dice are distinguishable, so (1, 6) and (6, 1) are different outcomes, giving six times six.

May I answer as a percentage?

Yes, 0.5 and 50 percent are both accepted, but keep one form throughout a single answer.

Must an expected frequency be a whole number?

No. Whole numbers are coincidence; an answer such as 2.45 is fine because it is an average, not a count.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.