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Teach Probability in four steps.

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

Gather a die, ₹1 and ₹2 coins, scrap paper, a pencil and an opaque bag.

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

Why it matters

  • A game's winning rule determines how many possible results win.
  • Shuffling cards makes the next card uncertain, but its chance can be calculated.
  • More entries in a fair prize draw give a greater chance of winning.

What to teach

Count outcomes and compare chances

Theoretical probability and complementary eventsCount matching outcomes out of all equally likely outcomes.Probability of drawing a cardCount qualifying cards out of all cards available to draw.Drawing an object at randomEach individual object has an equal chance of selection.Equally likely outcomesEach outcome has the same chance of happening.

How to teach it

Count outcomes and compare chances

A die beside its face list, with 5 and 6 circled.

1. Theoretical probability and complementary events

Count matching outcomes out of all equally likely outcomes.

Show
Turn a die to list its six faces. Circle 5 and 6: these win. Point to the uncircled numbers.
Ask
For a fair die, what is the chance of not winning?
Say
Four of six faces lose, so not winning has probability 46\frac{4}{6}.
Used in
Board-game players compare winning chances to choose between moves.
If your child says…“For not winning, I turn the winning fraction upside down.”
Ask
Do the winning and losing faces together fill the whole list?
What’s true
An event and its complement have probabilities that add to 1.
Four suit rows show all 52 cards, with the four aces outlined.

2. Probability of drawing a card

Count qualifying cards out of all cards available to draw.

Show
Look at the diagram’s suit rows. Count each row, then find the aces, jacks, queens and kings.
Ask
What is the chance of drawing an ace from this shuffled deck?
Say
Four of 52 equally likely cards are aces, so the chance is 113\frac{1}{13}.
Used in
Card players use the cards still available to judge their chances of a useful draw.
If your child says…“Aces are face cards too.”
Ask
Are the aces in the jack, queen or king columns?
What’s true
Each suit has three face cards: jack, queen and king.
An opaque bag beside three opened matching tickets marked A, A and B.

3. Drawing an object at random

Each individual object has an equal chance of selection.

Show
Mark matching paper tickets A, A and B. Fold them alike and mix thoroughly in an opaque bag. After the prediction, draw without looking and open all tickets.
Ask
What is the chance of drawing A?
Say
Two of the three equally likely tickets carry A, so its probability is 23\frac{2}{3}.
Used in
With one matching ticket per person, mixing and drawing without looking gives everyone equal chances.
If your child says…“There are two letters, so each has half the chance.”
Ask
How many separate tickets could give you A?
What’s true
Tickets have equal chances, but letters repeated on more tickets have greater chances.
₹1 and ₹2 coins beside four recorded pairs, with the ₹1 result first.

4. Equally likely outcomes

Each outcome has the same chance of happening.

Show
Place ₹1 left and ₹2 right. Keep ₹1 heads-up while turning ₹2 through heads and tails. Repeat with ₹1 tails-up. Record each pair in words, left coin first.
Ask
For fair coins tossed independently, what is the chance of exactly one head?
Say
Two of four equally likely outcomes give exactly one head, so its probability is 12\frac{1}{2}. Heads-tails and tails-heads count separately.
Used in
Games using several dice need complete outcome lists to compare winning rules fairly.
If your child says…“There are only three outcomes: two heads, one of each, or two tails.”
Ask
Can either coin supply the head when you get one of each?
What’s true
One of each includes two separate outcomes, depending on which coin shows heads.

Concept Lab · Common misconception

The tempting mistake in Probability

Open standalone misconception page →

Does a losing streak change the next fair toss?

Wrong

After three tails, heads is more likely on the next toss.

Why it feels right. I expect the short run to balance out.

Right

Independent fair tosses keep the same probabilities.

Worked example

Next toss · heads
TTT→H · 1/2
Next toss · tails
TTT→T · 1/2
After TTT, each next outcome still has the same probability for a fair coin.
  1. After TTT, the next toss still has two equally likely results.
  2. The probability of heads is ½.

Ask your child four quick questions.

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

  1. What must the chances of winning and not winning add to?

    1. Together they include every outcome, with none counted twice.

  2. How many face cards remain after all aces are removed?

    12. Removing aces leaves every jack, queen and king.

  3. Which letter is more likely to be drawn?

    A. A appears on two tickets, while B appears on one.

  4. What is the chance of heads on ₹1 and tails on ₹2?

    14\frac{1}{4}. This specifies one of the four equally likely ordered outcomes.

15 more questions with answers