Probability & Venn Diagrams
Explore sample spaces, calculate probabilities, use Venn diagrams to organise data, and understand complementary events.
Probability Basics
Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain).
P(event) = favourable outcomes ÷ total outcomes
The Probability Scale
A standard die is rolled. What is P(rolling a 3)?
Favourable outcomes: {3} → 1
Total outcomes: {1, 2, 3, 4, 5, 6} → 6
P(3) = 1/6
Sample Space
The sample space is the complete list of all possible outcomes for an experiment.
Coin flip
S = {Heads, Tails}
2 outcomes
Standard die
S = {1, 2, 3, 4, 5, 6}
6 outcomes
Venn Diagrams
A Venn diagram uses overlapping circles to show how different groups relate to each other. The overlap represents items in both groups.
Example: 30 students surveyed about Sports and Music
Sports only: 12 students
Music only: 6 students
Both: 8 students
Neither: 4 students
Total: 12 + 8 + 6 + 4 = 30 students. Total doing Sports = 12 + 8 = 20.
Complementary Events
The complement of an event is everything that is NOT that event. The probabilities of an event and its complement always add up to 1.
P(not A) = 1 − P(A)
Example: If P(rain) = 0.3, then P(no rain) = 1 − 0.3 = 0.7
Knowledge Check
Select the correct answer for each question. Questions get harder as you go!
Question 1
A bag has 3 red balls and 7 blue balls. What is the probability of picking a red ball?
Question 2
If the probability of winning a game is 0.4, what is the probability of NOT winning?
Question 3
How many outcomes are in the sample space when rolling a standard die?
Question 4
In a Venn diagram, 15 students play soccer, 10 play basketball, and 5 play both. How many play soccer only?
Question 5
A standard die is rolled. What is P(rolling an even number)?
Question 6
Using the Venn diagram: 30 students surveyed. Sports only = 12, Both = 8, Music only = 6, Neither = 4. What is P(a student does Music)?
Question 7
Two coins are flipped. What is the size of the sample space?
Question 8
In a class of 40: 22 like Maths, 18 like Science, 8 like both. How many like neither?
Question 9
A card is drawn from a standard 52-card deck. What is P(drawing a heart OR a king)?
Question 10
In a group of 50 people: 30 speak English, 25 speak Spanish, and 10 speak both. A person is chosen at random. What is P(they speak English only)?
Key Concepts Summary
- ●P(event) = favourable outcomes ÷ total outcomes (between 0 and 1).
- ●The sample space lists all possible outcomes of an experiment.
- ●Venn diagrams use overlapping circles to show relationships between groups.
- ●Complementary events: P(not A) = 1 − P(A).
- ●For “A or B”: add totals but subtract the overlap to avoid double-counting.