Chapter Overview
🧭Your visual roadmap through the chapter
Possible Outcomes
Tree diagrams, counting principle, permutations & combinations
Probability
Classical probability, addition & multiplication laws, conditional probability
Determining the number of possible outcomes for a situation (using tree diagrams, permutations and combinations), and measuring the likelihood that an event occurs (probability).
Estimating outcomes and likelihoods underpins prediction, risk assessment and decision-making in business situations.
Apply the counting principle, permutations and combinations; calculate probability using classical and relative-frequency approaches; apply addition, multiplication and conditional- probability rules; use probability tables and trees.
Key Definitions
📘Source-based terminology, exam-ready
Permutation
A set of several possible ways in which a number of things can be ordered or arranged — an "ordered combination" where order matters.
Key word: order matters
Combination
A group of items chosen from a larger number without regard for their order — order does NOT matter.
Key word: order doesn't matter
Factorial (n!)
An instruction to multiply all whole numbers counting down from n to 1 (e.g. 6!=6×5×4×3×2×1). Remember: 0! = 1.
Key word: 0! = 1
Probability
A measure of the likelihood of something happening; a number between 0 (cannot occur) and 1 (certain to occur).
Key word: 0 to 1
Outcome / Event / Trial
An outcome is what happens as a result of an event. An event is a group of one or more outcomes. A trial is a single attempt to obtain a given event.
Key word: outcome ⊆ event
Mutually Exclusive Events
Events where the occurrence of one prevents the occurrence of the other in a single observation — no common outcomes.
Key word: no overlap
Independent Events
Events where the outcome of one does not affect the outcome of the other.
Key word: no effect on each other
Conditional Probability
The probability of an event occurring given that another event has already occurred (one event's probability depends on another).
Key word: given that...
Core Concept
🎯Explained in exam-oriented language
CONCEPT → Possible Outcomes
MEANING
Counting how many ways a set of choices can be arranged or selected.
PURPOSE
Find the total number of arrangements or selections without listing every one.
APPLICATION
Product variations, team/committee selection, code/password combinations.
CONCEPT → Probability
MEANING
A number from 0 to 1 measuring how likely an event is to occur.
PURPOSE
Support predictions and decisions under uncertainty.
APPLICATION
Risk assessment and risk management in business.
Key Rules & Principles
📏Every source rule, exam-ready
RULE 1 Counting principle (multiplicative)
If there are m ways of doing one thing and n ways of doing another (independent), there are m × n ways of doing both
Extends to three or more events: m × n × p
RULE 3 Permutations with repeated items
Divide n! by x! for each group of identical/repeated items
With two repeated groups, divide by both xi! and xii!
RULE 4 Permutations vs Combinations
Use permutations (nPr) when the order of selection matters
Use combinations (nCr) when order does NOT matter
RULE 2 Additive principle
If m and n are the number of ways two mutually exclusive events can occur, the possibility that either occurs is m + n
Formula Zone — Counting, Permutations & Combinations
🧮Every formula supported by the source text
PERMUTATIONS (ALL N ITEMS)
n = number of items; arrange all of them, order matters.
PERMUTATIONS WITH IDENTICAL ITEMS
x = number of identical items/repetitions.
PERMUTATIONS OF SIZE R (NPR)
Choosing and arranging r items out of n, without repetition, order matters.
COMBINATIONS (NCR)
Also called the binomial coefficient. x=size of the group selected; order does not matter.
Formula Zone — Probability Laws
🧮Every formula supported by the source text
CLASSICAL PROBABILITY
Used when all outcomes are equally likely.
RELATIVE FREQUENCY
Used when outcomes are NOT all equally likely — based on observation/experiment.
ADDITION LAW (OVERLAPPING EVENTS)
Use when A and B can both occur together (not mutually exclusive).
ADDITION LAW (MUTUALLY EXCLUSIVE)
Use when A and B cannot both occur (no common outcomes).
MULTIPLICATION LAW (INDEPENDENT)
Use when one event's outcome does not affect the other.
MULTIPLICATION LAW (DEPENDENT) & CONDITIONAL PROBABILITY
P(B|A) = probability of B occurring given A has occurred.
COMPLEMENT RULE
The probability of an event plus the probability of it not occurring always equals 1. Useful for "at least one" questions.
Calculation Roadmap
🗺️General path for any probability question
1· Read Question
Identify events & what's asked
2· Identify Relationship
Exclusive/overlapping? Independent/dependent?
3· Select Formula
Addition, multiplication, or conditional law
4· Substitute
Put known probabilities into the formula
5· Calculate & Check
Solve, recheck, state final answer
Step-by-step Methods
🪜Kept separate — do not mix methods
METHOD A · Counting Principle / Tree Diagram
- 1List each independent choice and how many options it has
- 2Confirm the choices are independent of each other
- 3Multiply all the option counts together
- 4For mutually exclusive alternative routes, add instead
METHOD D · Basic Probability
- 1Check whether outcomes are equally likely
- 2If yes: P(event) = favourable ÷ total possible outcomes
- 3If not (based on observation): use relative frequency instead
- 4Express as a decimal or a percentage
METHOD B · Permutations
- 1Count total items (n) and, if relevant, the size chosen (r)
- 2Arranging all n items: use n!
- 3Some items identical: divide n! by x! (per repeated group)
- 4Choosing/arranging r of n: use n!÷(n−r)!
METHOD E · Addition Law (OR)
- 1Decide if events A and B can occur together (overlapping) or not (mutually exclusive)
- 2Mutually exclusive: P(A or B) = P(A)+P(B)
- 3Overlapping: subtract the shared outcomes, P(A or B)=P(A)+P(B)−P(A and B)
- 4State the final probability
METHOD C · Combinations
- 1Confirm order does NOT matter (a group/selection, not an arrangement)
- 2Identify n (total items) and x (size of the group selected)
- 3Substitute into nCx = n!÷[x!(n−x)!]
- 4For multi-category selections, multiply the separate nCx values together
METHOD F · Multiplication Law & Conditional Probability (AND)
- 1Decide if A and B are independent or dependent
- 2Independent (e.g. with replacement): P(A and B)=P(A)×P(B)
- 3Dependent (e.g. without replacement): P(A and B)=P(A)×P(B|A)
- 4For "at least one", use the complement: 1−P(none)
METHOD G · Multi-Stage Problems (Tables & Trees)
- 1Set up a probability table, contingency table, or probability tree for the stages/groups given
- 2Multiply along each branch/path for a joint probability
- 3Add across relevant branches/cells for a combined probability
- 4For a conditional question, divide the relevant joint probability by its row/column total
Different Cases / Scenarios
🔀Recognise the situation, pick the right route
IF choices are independent stages of one task
→ USE the counting principle: multiply the option counts
IF there are alternative, mutually exclusive routes
→ USE the additive principle: add the option counts
IF the order of selection matters
→ USE a permutation (n! or nPr)
IF the order of selection does NOT matter
→ USE a combination (nCx)
IF some items being arranged are identical
→ Divide n! by x! for each repeated group
IF events A and B can both happen (overlap)
→ P(A or B) = P(A)+P(B) − P(A and B)
IF one event's outcome does NOT affect the other
→ USE P(A and B)=P(A)×P(B) (independent)
IF a selection is made WITHOUT replacement
→ Events are dependent — use P(A)×P(B|A)
Comparison Tables
⚖️Don't confuse these!
| Permutation | Combination | |
|---|---|---|
| Order | Matters | Does not matter |
| Formula | n! or n!÷(n−r)! | n!÷[x!(n−x)!] |
| Example | Arranging books on a shelf in a given sequence | Choosing a committee/team from a group |
| Mutually Exclusive Events | Independent Events | |
|---|---|---|
| Meaning | Cannot both occur; no common outcomes | The outcome of one does not affect the other |
| OR rule | P(A or B) = P(A) + P(B) | N/A (this concerns AND, not OR) |
| AND rule | Cannot both occur, so P(A and B) = 0 | P(A and B) = P(A) × P(B) |
| Multi-stage Tool | What it is | Best used for |
|---|---|---|
| Probability table | Sets out probabilities of different combinations directly | A small number of categories/stages |
| Contingency table | Like a probability table, scaled up by a convenient total (e.g. 300 items) | Visualising outcomes as whole numbers of items |
| Probability tree | A diagram of branching links between possible probabilities | Multi-stage problems with sequential events |
Reading Tables & Trees
📌How to interpret the source's tools
Reading a probability tree
Each branch shows a probability; multiply along a path (branch × branch) for the joint probability of that path. Add the results of every path leading to the outcome you want. All final path probabilities must sum to 1.
Reading a contingency/two-way table
Rows and columns represent two variables (e.g. department × age group). A cell is the count/probability for that combination. Row and column totals give the marginal counts used for simple probabilities.
Conditional probability from a table
To find P(B|A), take the cell for "A and B" and divide it by A's row (or column) total — not the grand total.
Reading combined probability + counting problems
When a question needs both a count (nCx) and a probability, first find the total number of equally likely selections (nCx as the denominator), then count the favourable selections (as the numerator).
Worked Calculation Logic
✍️Method students must reproduce in the exam
GIVEN → FORMULA → SUBSTITUTION → CALCULATION → ANSWER
There are 4 white and 5 blue pens. What is the probability that two white pens are drawn respectively, if neither pen is replaced?
GIVEN: 4 white, 5 blue (9 total); draws are without replacement (dependent events)
FORMULA: P(A and B) = P(A) × P(B|A)
SUBSTITUTION: P(1st white) = 4/9; P(2nd white | 1st white) = 3/8
CALCULATION: P(A and B) = 4/9 × 3/8 = 12/72
FINAL ANSWER: P(both white) = 1/6 ≈ 0.167 or 16.7%
Exam Question Patterns
📝Counting — total arrangements/combinations for a scenario
Permutation/combination calculation — direct formula application
Basic probability — classical or relative-frequency calculation
Addition/multiplication law — OR / AND probability questions
Conditional probability — with/without replacement scenarios
Multi-stage problems using tables or probability trees
Exam Traps
⚠️⚠ WATCH OUT
- ✗ Permutation vs combination mix-up — check whether order genuinely matters before choosing the formula.
- ✗ Forgetting −P(A and B) when events overlap — only skip it for genuinely mutually exclusive events.
- ✗ Using P(A)×P(B) for dependent events instead of P(A)×P(B|A) — check for "without replacement".
- ✗ Forgetting 0! = 1 in factorial-based calculations.
- ✗ "At least one" questions — direct calculation is error-prone; use the complement 1−P(none) instead.
Quick Decision Tree
🌳Chapter 11 in 60 Seconds
⚡Counting: m×n (independent), m+n (exclusive)
nPr: n!/(n−r)! nCx: n!/[x!(n−x)!] P(event) = favourable/total
OR (overlap): P(A)+P(B)−P(A and B)
OR (exclusive): P(A)+P(B)
AND (independent): P(A)×P(B)
AND (dependent): P(A)×P(B|A)
Trap: "at least one" = 1−P(none)
Test yourself on this chapter
Practice 100 MCQs per chapter with answers – Chapter 1 is free for everyone. Enrol on WhatsApp for all 12 chapters, classes and tests on our student portal.