DSE Maths: probability and statistics foundations (tutor lesson plan)

2026-11-03

Tutor lesson-prep guides · Secondary Maths

Most students can compute an answer but cannot say where the denominator came from. In this lesson every denominator is counted, never recited.

Objectives — Students can: (1) write out the sample space of an experiment and count the outcomes (52 cards, 36 dice outcomes); (2) decide when to add and when to multiply, and explain mutually exclusive versus independent; (3) compute mean, median, mode and standard deviation from a set of five values and say what each statistic answers.

01 Prerequisites (3-minute check)

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

Skill	Quick check	Pass mark
Fractions	Reduce 12/52 to lowest terms	3/13
Averages	How the mean of four numbers is found	Total divided by the count
Square roots	Approximate the square root of 8.8	About 2.97

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Count the denominator	Start from 52 cards and two dice; write the full sample space	52 and 36 counted by the student
8-20 min	Add or multiply	One example of each rule; define mutually exclusive and independent	Name the rule before calculating
20-32 min	Expected value	Use the sum of two dice to reach 3.5	Say that 3.5 is never rolled
32-40 min	Five statistics	Use [4, 7, 7, 9, 13] for mean, median, mode and standard deviation	SD about 2.97
40-45 min	Exit check	Mark on the spot, correct on the spot	Three out of three

03 Teaching script (read it out)

Scene 1: Count the denominator before touching a formula

A deck has 52 cards in 4 suits of 13. For the probability of a heart, the denominator is the number of equally likely outcomes, 52, and the numerator is 13, so 13/52 = 0.25. Face cards are J, Q and K, which is 12 cards, so the probability is 12/52 about 0.23.
The denominator stays the same and the numerator changes, which is the point: the denominator is every outcome, not the outcome you want. Then move to two dice. There are 36 outcomes, not 12, because (1,2) and (2,1) are different. Have the student draw the 6 by 6 table before anyone says that a sum of 7 has six combinations.

Ask the student: why does rolling two dice give 36 outcomes rather than 12?

Scene 2: Addition is “or”, multiplication is “then”

The addition rule handles A or B: P(A union B) = P(A) + P(B) – P(A and B). For a heart or a face card, the heart face cards have been counted twice and must be subtracted once; only mutually exclusive events can be added without subtracting.
The multiplication rule handles A then B, and only independent events multiply directly. Two dice are independent because the first result does not change the second; two cards drawn without replacement are not, because the second draw faces a smaller deck.
Use one fixed question sequence: is this “or” or “then”, and do the events affect each other? Those two answers choose the rule.

Ask the student: when two cards are drawn without replacement, why is the second heart no longer 1/4?

Scene 3: The mean answers “typical”, the standard deviation answers “steady”

For [4, 7, 7, 9, 13]: the total is 40 over 5 values, so the mean is 8; the middle value is 7; the most frequent value is 7; the range is 9.
For the standard deviation, subtract the mean, square, average and take the root: the deviations are -4, -1, -1, 1 and 5, the squares sum to 44.0, the average is 8.8, and the root is about 2.97.
Each statistic answers a different question: the mean gives the typical value, the median resists extremes, the mode gives the most common value, and the standard deviation gives the spread. Students who say only that a bigger mean is better have never been told what each number is for.

Ask the student: if a value of 100 is added, which moves more, the mean or the median?

04 Question types and worked steps

Type	Example	Steps	Answer
Counting the sample space	Two dice are rolled; find the probability that the sum is 7	List or count combinations, write 36 as the denominator, count 6 outcomes	6/36 = 1/6
Addition rule	One card is drawn; find the probability that it is a heart or a face card	Add, subtract the overlap, state whether the events are mutually exclusive	13/52 + 12/52 – 3/52 = 22/52
Expected value	A fair die is rolled once; find the expected value	Multiply each outcome by its probability and add; say it need not be an outcome	3.5
Comparing spreads	Two data sets have the same mean; which is more consistent?	Compare range, then standard deviation, then explain in words	The smaller standard deviation is more consistent

Sanity check (teach this): Check the numbers and the clock: 52 cards, 4 suits of 13, 12 face cards; two dice give 36 outcomes and a sum of 7 has 6 of them, a probability of 0.167. For [4, 7, 7, 9, 13] the mean is 8, the median 7 and the standard deviation 2.97. Every figure can be counted or computed by the student, so nothing depends on a lookup table.

05 Independent practice (3 questions)

Question	Answer	What to watch
Two dice are rolled; find the probability that the sum is at least 10	6/36 = 1/6 (4+6, 5+5, 5+6, 6+4, 6+5, 6+6)	Whether 36 is used as the denominator and both (5,6) and (6,5) are counted
One card is drawn; find the probability that it is a heart or an ace	13/52 + 4/52 – 1/52 = 16/52 = 4/13	Whether the ace of hearts is subtracted once
For [4, 7, 7, 9, 13], find the mean and the standard deviation	Mean 8, standard deviation about 2.97	Whether the sum is divided by 5 rather than 4, and whether units are stated

06 Common mistakes and what to say

Mistake	Student writes	Say this
Wrong denominator	Answering 1/12 for a sum of 7 with two dice	“The total number of outcomes is 36, not 12. Draw the table first.”
Confusing exclusive with independent	“Two hearts is 1/4 times 1/4”	“The second draw has fewer cards and fewer hearts, so the events are not independent.”
Treating expected value as a prediction	“The expected value is 3.5, so the next roll must be 3 or 4”	“The expected value is a long-run average; no single roll equals 3.5.”
Dividing by n minus 1	Mixing the sample and population formulas	“In this syllabus divide by n unless the question says sample.”

07 Homework (paper, section and question numbers)

DSE maths probability paper — Probability questions 1 to 4: write the sample space and the number of outcomes before the numerator, in 20 minutes.
Statistics section of the same paper — Questions 5 to 7: compute mean, median, mode and standard deviation for [4, 7, 7, 9, 13], and write one sentence on what each statistic answers.

08 Exit check (last 5 minutes)

Question	Answer	Decision
Number of outcomes for two dice	36	Answering 12 means ordered pairs are not yet established
Expected value of one die	3.5	Answering 3 or 4 means expected value is still read as a prediction
Median of [4, 7, 7, 9, 13]	7	Answering 8 means median and mean are confused

09 Tutor FAQ

Must the standard deviation formula be memorised?

Yes, but only after the student can state the four steps: subtract the mean, square, average, take the root.

Can an expected value be a fraction or decimal?

Yes. It need not be a possible outcome, which is exactly what separates it from a probability.

When do mean and median differ most?

When there are extreme values: adding 100 pulls the mean up while the median barely moves.

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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.