Probability
Probability = favourable ÷ total. Always between 0 and 1. P(not A) = 1 − P(A).
A. Key Definitions
The Probability Scale — 0 to 1
Probability
A number from 0 to 1 that measures how likely an event is. 0 = impossible. 1 = certain.
FormulaP(Event) = Number of favourable outcomes ÷ Total possible outcomes
Sample Space (S)
The complete list of all possible outcomes of an experiment.
Example
Toss a coin → S = {H, T} (2 outcomes)
Roll a die → S = {1, 2, 3, 4, 5, 6} (6 outcomes)
Roll a die → S = {1, 2, 3, 4, 5, 6} (6 outcomes)
Favourable Outcomes
The outcomes that satisfy the event you are looking for.
Example
Event = rolling an even number on a die
Favourable = {2, 4, 6} → 3 outcomes
Favourable = {2, 4, 6} → 3 outcomes
Complementary Event
The complement of event A is "A does not happen". P(A) + P(not A) = 1.
FormulaP(not A) = 1 − P(A)
P(not getting 6) = 1 − 1/6 = 5/6
P(not getting 6) = 1 − 1/6 = 5/6
B. Basic Probability — Coins, Dice, Cards
Example 1 — Toss a coin. P(Heads)?
1Total outcomes = 2 (H or T)
2Favourable (Heads) = 1
P(Heads) = 1/2 = 0.5
Example 2 — Roll a die. P(even number)?
1Total outcomes = 6
2Even numbers = {2, 4, 6} → 3 favourable
P(even) = 3/6 = 1/2
Example 3 — Draw a card from 52. P(Ace)?
1Total = 52 cards. Aces = 4 (one per suit)
2P(Ace) = 4/52 = 1/13
P(Ace) = 1/13
C. Types of Events
Mutually Exclusive vs Independent Events
Mutually Exclusive Events
Two events that cannot happen at the same time. If one happens, the other cannot.
Formula & Example
P(A or B) = P(A) + P(B)
Rolling a 2 or a 5 on a die: P = 1/6 + 1/6 = 2/6 = 1/3
Rolling a 2 or a 5 on a die: P = 1/6 + 1/6 = 2/6 = 1/3
Independent Events
The outcome of one event does not affect the other. Tossing a coin twice — first toss doesn't change the second.
Formula & Example
P(A and B) = P(A) × P(B)
Two coins both land Heads: 1/2 × 1/2 = 1/4
Two coins both land Heads: 1/2 × 1/2 = 1/4
With vs Without Replacement
With replacement — put item back before second draw. Total stays the same.
Without replacement — don't put back. Total decreases by 1.
Without replacement — don't put back. Total decreases by 1.
Example — draw 2 cards
With replacement: P(Ace then King) = 4/52 × 4/52
Without replacement: P(Ace then King) = 4/52 × 4/51
Without replacement: P(Ace then King) = 4/52 × 4/51
Example — Bag has 3 red, 5 blue balls. Draw 2 without replacement. P(both red)?
1Total = 8. P(1st red) = 3/8
2After 1 red removed: 2 red left, 7 total. P(2nd red) = 2/7
3P(both red) = 3/8 × 2/7 = 6/56 = 3/28
P(both red) = 3/28
⚠ Common mistakes:
(1) Forgetting to subtract the overlap for non-mutually-exclusive events: P(A or B) = P(A)+P(B)−P(A∩B).
(2) Without replacement — the denominator decreases by 1 each draw.
(3) P values must always stay between 0 and 1 — if you get a negative or greater than 1, you made an error.
⚡ MCQ Tip: Mutually exclusive → add. Independent → multiply. Complement → 1 minus. These three rules cover 90% of probability MCQs.
Quick MCQ Revision
| Rule | Formula |
|---|---|
| Basic probability | Favourable ÷ Total |
| Complement | P(not A) = 1 − P(A) |
| Mutually exclusive (or) | P(A) + P(B) |
| Independent (and) | P(A) × P(B) |
| Range of probability | 0 (impossible) to 1 (certain) |
| 52-card deck | 4 suits × 13 cards each |