Which group has the greater chance on this fair die?

1–4. Four faces give that group, while only two give 5–6.

Is rolling 6 impossible because its chance is below halfway?

No. One face shows 6, so its chance is above zero.

What number goes underneath your sideways count, even with three landing types?

20. The count comes from 20 trials, not three possible landing types.

What total goes below four when calculating this chance?

6. All six faces can appear, including the four that qualify.

Why might the actual mango count differ from 8?

Only a sample was asked. The remaining pupils may have different preferences.

After nine tails, what is the next head’s chance with this fair coin?

1/2. Earlier tosses do not change the coin’s next independent result.

How many listed arrangements have at least one head?

Three. HH qualifies alongside HT and TH.

What must match before we can give every path the same chance?

Their branch probability products. Counting paths alone cannot establish equal chances.

Teach The Mathematics of Maybe: Introduction to Probability in eight steps.

For each idea: show it, ask before you tell, then say one line.

Gather a die, steel cup, paper cup, two different coins, paper and pencil.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

Rain forecasts express uncertainty, helping families decide whether to carry umbrellas.
Games use chance to keep their results uncertain.
School fruit surveys help estimate how much fruit to buy.

What to teach

How to teach it

Describe and measure chance

1. Randomness

We know the possibilities, but cannot be certain what comes next.

Show: Inspect a fair die. Group its faces as 1–4 and 5–6. Shake it inside a steel cup. Roll after the prediction.
Ask: Which group do you predict will appear on the next roll?
Say: The next group stays uncertain, but four faces favour 1–4.
Used in: Shuffling playing cards keeps the next card uncertain, making games unpredictable.

If your child says…“There are two groups, so their chances match.”

Ask: How many faces belong to each group?
What’s true: Uncertain results need not have equal chances.

Probability diagram from impossible to certain, with even chance halfway.

2. The Probability Scale

Chance ranges from impossible at zero to certain at one.

Show: Find zero, halfway and one on the diagram. Notice the chance labels between them. Inspect every die face.
Ask: Where will the chance of rolling 6 sit on this line?
Say: One face qualifies, so mark $\frac{1}{6}$ : less likely still allows a 6.
Used in: Rain probabilities help families plan outings while allowing for uncertain weather.

If your child says…“Less likely means it cannot happen.”

Ask: Can you find a face showing 6?
What’s true: A small chance is still above zero.

Find and use probabilities

A sideways paper cup beside 20 landing records, with sideways records circled.

3. Experimental Probability and Relative Frequency

Divide an event’s recorded count by the total trials.

Show: After the prediction, toss the paper cup onto a clear floor 20 times. Record upright, upside down or sideways. Circle the sideways records.
Ask: What fraction of this cup’s landings do you predict will be sideways?
Say: The circled count is the frequency. Divide it by 20 for relative frequency, your experimental probability.
Used in: Recorded bus delays help travellers estimate how often a bus arrives late.

If your child says…“I divide by three because there are three landing types.”

Ask: How many tosses gave you those records?
What’s true: The sideways count belongs to all 20 tosses.

$\text{Qualifying outcomes}=4$

$\text{Total outcomes}=6$

4. Theoretical Probability

With equal chances, divide matching outcomes by all outcomes.

Show: Write these lines. Say: “Each face of this fair die has the same chance.”
Ask: Before any rolls, what chance does rolling 1, 2, 3 or 4 have?
Say: Four faces qualify among six equally likely faces. The chance is $\frac{4}{6}=\frac{2}{3}$ .
Used in: Raffle organisers count equally likely tickets to work out each entry’s winning chance.

If your child says…“I divide four matching faces by the two others.”

Ask: How many faces could appear, including those that qualify?
What’s true: The total includes matching faces and the other faces.

$\text{Sample share}=\frac{2}{5}$

$\text{Estimated count}=\frac{2}{5}\times20$

5. Statistical Data and Sampling

Use a representative sample’s share to estimate the whole group’s count.

Show: Write this line. Say: “In this practice survey, 2 of 5 randomly selected pupils chose mango. There are 20 pupils altogether.”
Ask: From this survey, how many of the 20 pupils would you expect to choose mango?
Say: Four lots of two give an estimate of 8 pupils choosing mango.
Used in: Canteens use preference surveys to estimate food orders and reduce leftovers.

If your child says…“Exactly 8 pupils will want mango.”

Ask: Have we asked every pupil?
What’s true: A sample gives an estimate; the actual count may differ.

Nine recorded tails beside two equally likely branches for the next fair toss.

6. Experimental and Theoretical Probability; Gambler’s Fallacy

Earlier independent results cannot change the next trial’s chance.

Show: Look at the diagram’s example record: nine tails from independent tosses of a fair coin. Find the recorded heads fraction, then the next toss’s branches.
Ask: After these nine tails, is heads more likely than tails on the next toss?
Say: No heads appeared in this record, but independent tosses do not change each other’s chances. The next heads chance is still one-half.
Used in: Players avoid unreliable predictions by knowing earlier fair rolls cannot make a result due.

If your child says…“Heads must catch up now.”

Ask: Which branch would change because earlier tosses were tails?
What’s true: Short runs can differ from half without changing the next toss’s chance.

List outcomes and follow paths

Two different coins beside a sheet listing HH, HT, TH and TT.

7. Sample Space and Events

List all possible results; collect matching results into an event.

Show: Set two different coins left and right. Turn them to make HH, HT, TH and TT. Write each once: H means heads, T means tails.
Ask: Which arrangements will count for ‘at least one head’?
Say: Circle HH, HT and TH: ‘at least one’ includes both heads. The complete list is the sample space; the circled outcomes form this event.
Used in: Games group several winning results into one event, making winning chances easier to count.

If your child says…“At least one head means exactly one.”

Ask: Does HH contain at least one head?
What’s true: Both heads qualify because one head is the minimum.

Tree of two fair tosses, with half-chance branches and the HH path highlighted.

8. Tree Diagrams

Multiply branch chances to find each complete path’s chance.

Show: Find the four paths in the diagram for independent fair tosses. Follow HH and find both branch chances. P(HH) means the chance of heads twice.
Ask: Looking at HH’s two half-chance branches, what chance do you predict for heads twice?
Say: Half of a half gives $P(HH)=\frac{1}{4}$ .
Used in: Menus pair snacks with drinks to list every combination.

If your child says…“Four paths always means a quarter for each.”

Ask: What would change if one branch had a different chance?
What’s true: Paths have equal chances only when their branch products match.

A real problem

A sample survey estimated Bengal’s jute production.

Mahalanobis’s large sample survey studied jute growing area and yield in 1941.

It covered Bengal province as a whole.

Its production estimate closely matched the figure from customs and trade.

Indian Statistical Institute / Google Arts & Culture The New Indian Express

Plain jute sacks stacked on a grain market floor.

Concept Lab · Common misconception

The tempting mistake in The Mathematics of Maybe: Introduction to Probability

Open standalone misconception page →

Is every named outcome equally likely?

Wrong

A spinner with red and blue has a ½ chance of red.

Why it feels right. There are two colour names.

Right

For a fair spinner, the size of each sector matters.

Worked example

Red occupies three quarters of the spinner. The blue quarter is a smaller target.

Make red occupy three quarters of the circle and blue one quarter.
The red probability is ¾.

Ask your child eight quick questions.

One per step. The answer and the reason are given.

Which group has the greater chance on this fair die?

1–4. Four faces give that group, while only two give 5–6.
Is rolling 6 impossible because its chance is below halfway?

No. One face shows 6, so its chance is above zero.
What number goes underneath your sideways count, even with three landing types?

20. The count comes from 20 trials, not three possible landing types.
What total goes below four when calculating this chance?

6. All six faces can appear, including the four that qualify.
Why might the actual mango count differ from 8?

Only a sample was asked. The remaining pupils may have different preferences.
After nine tails, what is the next head’s chance with this fair coin?

$\frac{1}{2}$ . Earlier tosses do not change the coin’s next independent result.
How many listed arrangements have at least one head?

Three. HH qualifies alongside HT and TH.
What must match before we can give every path the same chance?

Their branch probability products. Counting paths alone cannot establish equal chances.

15 more questions with answers

Teach The Mathematics of Maybe: Introduction to Probability in eight steps.

Why it matters