I need to tell you something that might sting. Probability and counting principle is worth 15 marks in Paper 1. Fifteen marks that most students walk away from because they decided it was "too confusing" somewhere in Grade 11 and never looked back. Meanwhile, the students who actually learn the rules pick up those 15 marks in under 20 minutes of exam time. That is 15 marks for 20 minutes of work. There is no better deal in the entire NSC exam. Stop avoiding this topic. It is easier than you think.
In This Post You Will Learn
✓ The fundamental counting principle and how to use it for arrangement problems
✓ How factorial notation works and when to use it
✓ The difference between permutations and combinations (and when each applies)
✓ How to calculate probability using Venn diagrams, tables, and tree diagrams
✓ What mutually exclusive and independent events mean and how to test for them
✓ How to handle "at least one" problems without losing your mind
The Fundamental Counting Principle
This is the foundation of everything in this topic. Get this right and the rest falls into place.
If you have m ways to do the first thing and n ways to do the second thing, then you have m x n ways to do both things.
That is it. Multiply the choices.
Example: You have 4 shirts and 3 pants. How many different outfits can you make?
4 x 3 = 12 outfits.
Simple. But the NSC takes this further.
Arrangement Problems With Restrictions
Example: How many 4-digit codes can be made using the digits 1 to 9 if no digit may be repeated?
Think about each position:
Position 1: 9 choices (any digit from 1 to 9)
Position 2: 8 choices (one digit used up)
Position 3: 7 choices
Position 4: 6 choices
Total = 9 x 8 x 7 x 6 = 3 024 codesNow add a restriction: The code must start with an even number.
Even digits from 1 to 9: 2, 4, 6, 8. That is 4 options.
Position 1: 4 choices (must be even)
Position 2: 8 choices (one digit used)
Position 3: 7 choices
Position 4: 6 choices
Total = 4 x 8 x 7 x 6 = 1 344 codesPro tip from Mr Sawaya: Always deal with the restricted position FIRST. If the question says "must start with" or "must end with" or "the vowels must be together," handle that restriction before anything else. Then fill in the remaining positions.
Factorial Notation
The symbol n! (read as "n factorial") means multiply all positive integers from n down to 1.
5! = 5 x 4 x 3 x 2 x 1 = 120
4! = 4 x 3 x 2 x 1 = 24
3! = 3 x 2 x 1 = 6
2! = 2 x 1 = 2
1! = 1
0! = 1 (this is a rule, just accept it)Why do we need this? Because the number of ways to arrange n different objects in a row is n!
Example: How many ways can 6 people sit in a row?
6! = 720 ways.
Arrangements With Identical Items
If some items are identical, you divide by the factorial of each repeated group.
Example: How many different arrangements can be made from the letters of the word COFFEE?
COFFEE has 6 letters: C, O, F, F, E, E
F appears twice. E appears twice.
Total arrangements = 6! / (2! x 2!)
= 720 / (2 x 2)
= 720 / 4
= 180Items That Must Stay Together
If certain items must be together, treat them as one unit. Then arrange the units. Then arrange the items within that unit.
Example: 5 friends sit in a row. Two of them (A and B) must sit next to each other.
Step 1: Treat A and B as one unit. Now you have 4 units to arrange.
4! = 24
Step 2: A and B can swap within their unit.
2! = 2
Step 3: Total = 24 x 2 = 48
Probability Rules You Must Know
Here are the core rules. These are not optional. You need all of them.
| Rule | Formula | When to Use |
|------------------------|--------------------------------------|-------------------------------------|
| Basic probability | P(A) = n(A) / n(S) | Always the starting point |
| Complement | P(not A) = 1 - P(A) | "At least one" or "not" questions |
| Addition (OR) | P(A or B) = P(A) + P(B) - P(A and B)| When events can overlap |
| Mutually exclusive | P(A and B) = 0 | Events cannot happen together |
| Independent events | P(A and B) = P(A) x P(B) | One event does not affect the other |Mutually Exclusive vs Independent
These two concepts are tested every year. Know the difference.
Mutually exclusive: Events cannot happen at the same time. If one happens, the other cannot.
Example: Rolling a die and getting a 3 AND a 5 on the same roll. Impossible. So P(3 and 5) = 0.
Test: P(A and B) = 0
Independent: One event does not affect the probability of the other.
Example: Flipping a coin and rolling a die. The coin result has no effect on the die result.
Test: P(A and B) = P(A) x P(B)
Critical exam trap: Mutually exclusive events are NEVER independent (unless one of them has probability 0). Students mix these up constantly. If two events are mutually exclusive, knowing that one happened tells you the other definitely did not happen. That means they affect each other. That means they are dependent.
Venn Diagrams: The Visual Tool
Venn diagrams show overlapping events. The exam gives you information and you fill in the diagram.
┌─────────────────────────────────────┐
│ S (sample space) │
│ │
│ ┌──────────┬──────────┐ │
│ │ │ │ │
│ │ A only │ A and B │ B only │
│ │ │ │ │
│ └──────────┴──────────┘ │
│ │
│ Neither A nor B │
└─────────────────────────────────────┘The four regions must add up to 1 (or to the total number of outcomes).
Golden rule for filling in Venn diagrams: Start with the OVERLAP (A and B) first. Then subtract to find the "only A" and "only B" regions. Then use the total to find "neither."
Worked Example: Venn Diagram Problem
In a class of 40 students, 25 play soccer, 18 play cricket, and 10 play both.
Step 1: Fill in the overlap first.
A and B = 10
Step 2: Soccer only = 25 - 10 = 15
Step 3: Cricket only = 18 - 10 = 8
Step 4: Neither = 40 - 15 - 10 - 8 = 7
┌─────────────────────────────────┐
│ S = 40 │
│ ┌──────────┬──────────┐ │
│ │ │ │ │
│ │ 15 │ 10 │ 8 │
│ │ Soccer │ Both │Cricket│
│ │ only │ │ only │
│ └──────────┴──────────┘ │
│ 7 (neither) │
└─────────────────────────────────┘P(soccer or cricket) = (15 + 10 + 8) / 40 = 33/40
P(neither) = 7/40
For full live lessons on this topic, see our Grade 12 Maths tuition page.
The "At Least One" Trick
Whenever a question says "at least one," do NOT try to calculate it directly. Use the complement.
P(at least one) = 1 - P(none)
This is faster, cleaner, and avoids errors every time.
Example: A coin is flipped 4 times. What is the probability of getting at least one head?
P(no heads) = P(all tails) = (1/2)⁴ = 1/16
P(at least one head) = 1 - 1/16 = 15/16
If you tried to calculate P(exactly 1 head) + P(exactly 2) + P(exactly 3) + P(exactly 4), you would need four separate calculations. The complement method gives you the answer in two lines.
Contingency Tables (Two-Way Tables)
The NSC sometimes gives you data in a table and asks probability questions.
Example:
| | Passed | Failed | Total |
|-------------|--------|--------|-------|
| Studied | 45 | 5 | 50 |
| Did not study| 15 | 35 | 50 |
| Total | 60 | 40 | 100 |P(passed) = 60/100 = 0.6
P(studied and passed) = 45/100 = 0.45
P(passed, given that they studied) = 45/50 = 0.9
That last one is conditional probability. "Given that they studied" means your sample space shrinks to only the students who studied (50 students), not all 100.
Testing for Independence Using a Table
Are "studying" and "passing" independent?
Check: Does P(studied and passed) = P(studied) x P(passed)?
P(studied and passed) = 45/100 = 0.45
P(studied) x P(passed) = 50/100 x 60/100 = 0.5 x 0.6 = 0.30
0.45 ≠ 0.30, so the events are NOT independent.
This makes sense. Studying does affect your chance of passing. They are dependent events.
If you need to brush up on your graph reading skills, check out our guide on Grade 12 Functions and Graphs - Everything You Need to Know.
Common Mistakes Students Make
- Not dealing with the restricted position first in counting problems
If the question says "the code must start with a vowel," fill in position 1 first (with the number of vowels), then fill in the remaining positions. If you fill in position 1 last, you might overcount or undercount because some digits are already used up and you lose track.
- Forgetting to divide by repeated items
The word MISSISSIPPI has 11 letters but many repeats: S appears 4 times, I appears 4 times, P appears 2 times. The answer is 11! / (4! x 4! x 2!), not just 11!. Leaving out the denominators gives an answer that is wildly too large.
- Adding probabilities when you should multiply
P(A or B) uses addition. P(A and B) uses multiplication (if independent). Students mix up "or" and "and." Remember: "or" means at least one of them happens (add). "And" means both happen (multiply, if independent).
- Using the wrong sample space for conditional probability
When a question says "given that event A has occurred," the sample space is no longer the full set. It shrinks to only the outcomes where A happened. Students use the full sample space and get the wrong answer. Look for the words "given that," "if we know that," or "among those who" as signals for conditional probability.
- Confusing mutually exclusive with independent
If P(A and B) = 0, the events are mutually exclusive. But that does NOT make them independent. Independent means P(A and B) = P(A) x P(B). You must check the correct condition for whichever one the question asks about.
How This Topic Appears in the NSC Exam
Probability and counting principle appears in Paper 1 of the NSC Maths exam.
It typically carries around 15 marks and appears towards the end of the paper, usually as Question 9 or Question 10.
The DBE splits this into two parts most years:
Part 1 is a counting principle question (5 to 7 marks). It asks how many arrangements, codes, or selections are possible, usually with some restriction. Recent exams have included questions about arranging letters with conditions (vowels together, must start with a certain letter) and creating number codes with or without repetition.
Part 2 is a probability question (8 to 10 marks). It may involve a Venn diagram, a contingency table, or a tree diagram. You will almost always be asked to determine whether events are independent or mutually exclusive. The "show that" format is common: "Show that events A and B are independent."
In the 2023 NSC exam, this topic included a counting question about arranging letters of a word with repeated letters and a restriction, followed by a probability question using a two-way table where students had to test for independence.
Typical mark split:
| Sub-topic | Marks |
|-----------------------|--------|
| Counting principle | 5-7 |
| Probability (Venn/table)| 8-10 |
| TOTAL | ~15 |The counting principle part is usually the easier half. If you can do the fundamental counting principle and factorials, that is 5 to 7 marks you can secure quickly. The probability half requires more thought but follows predictable patterns. Practise 3 to 4 past paper questions and you will see every variation.
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