Probability Using Sets and Venn Diagrams

Lesson 4 of 4: Sets

In this lesson:

• Express probability as a fraction between 0 and 1
• Calculate for simple events
• Apply the complement rule:
• Read probabilities directly from a Venn diagram

What You Will Learn This Lesson

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

1. Explain probability as a fraction between 0 and 1
2. Calculate for simple events
3. Apply the complement rule
4. Read probabilities from a completed Venn diagram

Which Outcome Is More Likely Here?

Esther picks one mango at random from a bag:

• 4 ripe mangoes (she wants one of these)
• 6 unripe mangoes

Show of hands: who thinks she will get a ripe one?

How would you express "how likely" as a number?

Probability Expressed as a Fraction

• = number of favourable outcomes (what you want)
• = total number of equally likely outcomes

Worked Example: Rolling a Die

,

Always simplify the fraction where possible.

Mango Bag: Compute These Yourself

Esther's bag: 4 ripe, 6 unripe — 10 mangoes total,

Now you compute:

Answers: P(unripe) = 6/10 = 3/5; P(ripe or unripe) = 10/10 = 1

The Boundary Values Zero and One

Rule: always. If your answer is , recheck .

Quick Check: Spinner One to Eight

A spinner has equal sections numbered 1 to 8. One spin.

Answers: 1/8; 4/8 = 1/2; 2/8 = 1/4

Sample Space = Universal Set

The sample space is every possible outcome. An event is a subset of :

• for a standard die
• = "an even number"

Same , as the Venn lessons.

Writing in Set Notation

The shaded region contains the favourable outcomes.
The unshaded region contains the unfavourable outcomes.

The Complement Rule for Probability

Every outcome is in or in — no third option.

Therefore:

Why? When is large, complement is faster than counting.

Worked Example: Complement of a Prime

Spinner with sections 1–8. .

Worked Example: Complement of a Multiple

Spinner with sections 1–8. .

Predict First, Then Verify Below

Spinner with sections 1–8. You computed .

Before using the complement rule, predict:

What is ?

• A.
• B.
• C.

Commit to your answer, then verify using .

Quick Check — Complement Rule

Compute for each event using :

1. A bag has 3 red and 7 blue counters. . What is ?
2. A class has 30 learners; 18 are girls. . What is ?

Answers: 7/10; 2/5

From Region Counts to Probabilities

The Venn diagram from Lesson 3: , = even, = multiples of 3

Region counts (from Lesson 3):

What single operation converts every count to a probability?

Venn Diagram as a Probability Calculator

Reading Probabilities from the Diagram

From the probability diagram, :

Verify: Probabilities Sum to 1

Probability version of the Lesson 3 sum check:
counts sum to ; probabilities sum to .

Worked Example: Novels and Comics

20 learners: 12 novels, 9 comics, 5 both.

Draw the diagram, intersection first:

• Novels only
• Comics only
• Neither

Reading Probabilities: Novels and Comics

. Divide every count by 20.

Check: ✓

Your Turn — Football and Netball

30 learners: football, netball, .

1. Draw the diagram; fill four region counts.
2. Check counts sum to 30.
3. Find each region's probability.
4. Check probabilities sum to 1.

Answers: football only = 18−7 = 11; netball only = 14−7 = 7; both = 7; neither = 30−(11+7+7) = 5. Check: 11+7+7+5 = 30 ✓. Probabilities: 11/30, 7/30, 7/30, 5/30 = 1/6.

Watch Out — Common Errors

Error 1:

• ✗ — wrong
• ✓ If , recount

Error 2: not dividing by

• ✗ "", not
• ✓ Counts probabilities

Error 3: vs

• ✗ Centre count alone for
• ✓ Union = three regions; intersection = one

Key Takeaways From This Lesson

✓ , always
✓ Complement rule:
✓ Venn → probabilities: divide every region count by
✓ Sum check: four region probabilities sum to

Sets topic complete. Term 2 Data Handling extends this — same formula.