Cambridge Lower Secondary · Grade 7 · Probability & Statistics
Sample Space
Two coins: {HH, HT, TH, TT} All possible outcomes listed systematically
Combined Probability
P(double 6 on two dice) = 1/36 Count favourable ÷ total outcomes in grid
Complementary Events
P(not A) = 1 − P(A) Everything that isn't event A
Quick probability spinner — see the sample space live! 🌀
Click to spin!
Sample space: {Red, Blue, Green, Yellow}
What you'll learn:
Listing all possible outcomes (sample space) systematically
Drawing sample space diagrams (2D grids) for two combined events
Calculating P(A and B) from a sample space grid
Mutually exclusive events: P(A or B) = P(A) + P(B)
Complementary events: P(not A) = 1 − P(A)
Using tree diagrams to show combined outcomes
Solving problems using Venn diagrams
📖 Learn: Sample Space & Combined Probability
Part 1: What is a Sample Space?
The sample space is the set of all possible outcomes of an experiment. We list them using:
📋 A list — e.g. rolling one die: {1, 2, 3, 4, 5, 6}
🌳 A tree diagram — shows branching outcomes for multiple events
⊞ A sample space diagram (grid) — for two combined events
💡 The total number of outcomes in the sample space is used as the denominator when calculating probability.
Example: Flipping two coins — sample space: {HH, HT, TH, TT} — 4 outcomes total
P(exactly one Head) = 2/4 = 1/2 (HT and TH are favourable)
Part 2: Sample Space Diagrams (2D Grids)
When we combine two independent events (e.g. rolling two dice), we draw a 2D grid. The row headings are outcomes for Event 1, column headings for Event 2.
🎲 Rolling two dice: each axis runs 1–6, giving 6 × 6 = 36 total outcomes
Die 1 \ Die 2
1
2
3
4
5
6
1
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(1,6)
2
(2,1)
(2,2)
(2,3)
(2,4)
(2,5)
(2,6)
3
(3,1)
(3,2)
(3,3)
(3,4)
(3,5)
(3,6)
4
(4,1)
(4,2)
(4,3)
(4,4)
(4,5)
(4,6)
5
(5,1)
(5,2)
(5,3)
(5,4)
(5,5)
(5,6)
6
(6,1)
(6,2)
(6,3)
(6,4)
(6,5)
(6,6)
🟡 Yellow = "at least one 6". Green = "double 6". P(double 6) = 1/36.
Part 3: Combined Probability from a Sample Space
To find P(A and B), count the cells in the grid that satisfy both conditions.
📌 Formula: P(event) = number of favourable outcomes ÷ total outcomes
Example: P(sum = 7) on two dice — count pairs that add to 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → 6 outcomes
P(sum = 7) = 6/36 = 1/6
💡 Systematic listing avoids missing outcomes. Always check your total equals the full sample space size.
Part 4: Mutually Exclusive Events
Two events are mutually exclusive if they cannot both happen at the same time.
📌 Rule: P(A or B) = P(A) + P(B) (only when A and B are mutually exclusive)
Example: Rolling a die — A = "get a 3", B = "get a 5". Can't roll both at once → mutually exclusive.
P(A or B) = P(3) + P(5) = 1/6 + 1/6 = 2/6 = 1/3
⚠️ If events CAN overlap (e.g. "even number" and "greater than 4"), they are NOT mutually exclusive and you cannot simply add the probabilities.
Part 5: Complementary Events
The complement of event A (written A' or "not A") is every outcome that is NOT in A.
📌 Rule: P(not A) = 1 − P(A)
Example: P(rolling a 6) = 1/6, so P(not rolling a 6) = 1 − 1/6 = 5/6