Probability of Numbers

Lesson 4 of 4: Data Handling

In this lesson:

• Use P(A) = n(A)/n(E) for numerical outcomes
• Define events by number properties: prime, even, multiple
• Express probability as a fraction, decimal, or percentage

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Use for numerical outcomes
2. Define events by number properties and list favourable outcomes
3. Express probability as a fraction, decimal, and percentage
4. Solve problems using the complement rule

You Roll a Fair Die Once...

What is the chance it lands on a prime number?

• The die has six faces: 1, 2, 3, 4, 5, 6
• You know what prime means from patterns-02
• You know the probability formula from sets-04

Can you almost answer this? Think for a moment.

Recall: Which Outcomes Count as Favourable?

Favourable = on die and matches event.

The Probability Formula You Already Know

Complement rule: · Impossible = 0, certain = 1

Our Sample Space: The Die

Quick Check: Formula in Action

Use the die with , :

1. What is ?
2. What is ? — explain why
3. What is ?

Write your answers before the next slide.

Events Defined by Number Properties

Many questions ask about a property, not a specific value.

Examples: even, prime, multiple of 3, greater than 4

Method:

1. List numbers in with the property
2. Count them → , then divide by

The listing step forces you to check each number.

Worked Example: P(even) on a Die

Event: rolling an even number on our die,

Step 1 — List even numbers in :

Step 2 — Count:

Step 3 — Divide:

Worked Example: P(multiple of 3)

Event: rolling a multiple of 3,

Step 1 — List multiples of 3 in :

Step 2 — Count:

Step 3 — Divide:

Worked Example: P(greater than 4)

Is each number in strictly greater than 4?

• 4: No — 4 is not greater than 4 (excluded)
• 5: ✓ Yes
• 6: ✓ Yes

, so

"Greater than 4" excludes 4 itself.

Using the Complement to Save Counting

Complement rule:

Example: on a die

Predict: List the Primes on Our Die

The die has sample space .

Predict: List the prime numbers in this set:

Write your list now. Do not advance until you have committed.

Answer: Primes on the Die

Answer: Primes in are

• 1 is NOT prime — a prime must have exactly two different factors. The factors of 1 are: just {1}. Only one factor. Not prime.
• So — not

One Probability, Three Ways to Write It

These are not three different answers — three ways of writing the same probability.

• The probability is one number
• The question tells you which form to give
• Converting between forms is just notation, not a change in meaning

The Recipe: Fraction to Decimal to Percentage

Step 1: Divide numerator by denominator → decimal

Step 2: Multiply by 100 → percentage

Anchor check: Certain = | Impossible =

All Three Forms: P(even) on a Die

Converting a Repeating Decimal: P(multiple of 3)

Anchor check: ✓ and ✓

Your Turn: Convert P(greater than 4)

From the earlier slide:

1. Convert to a decimal (divide top by bottom)
2. Convert your decimal to a percentage (× 100)

Write both answers before advancing.

Answers: P(greater than 4) Converted

Answer:

• Decimal:
• Percentage:

What If the Sample Space Changes?

A spinner has the numbers 1 to 10. You spin once.

,

Find: as a fraction, decimal, and percentage.

Step 1: List the primes in {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} — remember, 1 is not prime.

Spinner Primes: Larger Sample Space

Primes: , so

Five Steps That Handle Any Probability Problem

• List the favourable numbers (by property)
• Count them →
• Divide by → fraction
• Convert if asked → decimal (÷) → percentage (×100)
• Complement if "not A" is easier →

Works on any sample space — die, spinner, cards, numbered tickets.

Worked Example: P(multiple of 5) from 1–20

Cards 1–20, . List: ,

Complement:

Find the Error: Who Is Prime?

A student wrote: "Primes in are , so ."

Your task:

1. Find the error in the student's list
2. Write the correct list and probability
3. Explain in one sentence why the error happened

No scaffolding — complete all three parts before advancing.

Error Found: 1 Is Not Prime

Correction:

• Error: 1 was included as prime — 1 has only one factor (itself), not two
• Correct list: , so
• Correct probability:

Guided Practice: Two Conditions at Once

Die: . Find .

Numbers must satisfy both conditions.

Step 1: List even numbers:

Step 2: Keep only those less than 5:

Step 3: Count and divide.*

Answer: Even AND Less Than Five

Answer:

• Even numbers:
• Even AND less than 5: (6 is not less than 5)

Practice: Find P(factor of 12)

Cards numbered 1 to 20, one drawn at random.

Find: as a fraction and as a percentage.

Factors of 12 are numbers that divide 12 exactly.

Work through all steps: list, count, divide, convert.

Answer: P(factor of 12) = 30%

Factors of 12 are — six factors.

Lesson Summary: Key Takeaways and Watchouts

• List → count → divide → convert; complement = 1 − P(A)
• Fraction = decimal = percentage — same probability
• 1 is not prime; die primes: 2, 3, 5
• "Greater than 4" excludes 4; favourable: 5, 6
• ×100: 0.2 × 100 = 20%, not 2%

This Lesson Closes Term 2 Data Handling

In the PLE examination, probability questions appear on:

• Numbered spinners and dice
• Coloured balls in a bag
• Drawing cards from a numbered set

The structure is always .

You have the method. The PLE asks you to apply it in any form.