Class 9 · Mathematics · Probability
🎲 The Mathematics of Maybe
Introduction to Probability — Complete notes with experiments, formulas, diagrams, and exam practice
Experiment
Sample Space
Event
Empirical Probability
Theoretical Probability
Complementary Events
Impossible & Sure Events
📋 Table of Contents
Introduction — What is Probability?
Every day we deal with uncertainty. Will it rain today? Will I get a 6 when I throw a die? Will heads come up when I flip a coin? Probability is the branch of mathematics that helps us measure how likely an event is to happen.
Real-Life Probability Around You
Weather forecasters say “70% chance of rain” • Doctors estimate “1 in 1000 risk of side effects” • Cricket commentators discuss chances of a team winning • Insurance companies calculate risk. All of this is probability in action!
Weather forecasters say “70% chance of rain” • Doctors estimate “1 in 1000 risk of side effects” • Cricket commentators discuss chances of a team winning • Insurance companies calculate risk. All of this is probability in action!
Probability always lies between 0 (impossible) and 1 (certain)
0 ≤ P(E) ≤ 1
The probability of any event is always a number between 0 and 1 (inclusive).
Key Terms & Definitions
| Term | Meaning | Example |
|---|---|---|
| Random Experiment | An action whose outcome cannot be predicted in advance | Tossing a coin, rolling a die |
| Trial | Each performance of the random experiment | Tossing the coin once |
| Outcome | A possible result of one trial | Getting Heads |
| Sample Space (S) | The set of ALL possible outcomes | S = {H, T} for a coin |
| Event (E) | A specific outcome or set of outcomes we are interested in | E = getting Heads |
| Favourable Outcomes | Outcomes that satisfy the condition of the event | Only {H} favours “getting Heads” |
| Equally Likely Outcomes | All outcomes have the same chance of occurring | Each face of a fair die has equal chance |
| Sure / Certain Event | Always happens; P = 1 | Getting a number ≤ 6 when rolling a die |
| Impossible Event | Never happens; P = 0 | Getting 7 on a normal die |
| Complementary Event | Event “NOT E”; everything except E | If E = Heads, then Ē = Tails |
🧩 Sample Space Examples
🪙 Tossing 1 Coin
S = {H, T}
Total outcomes = 2
S = {H, T}
Total outcomes = 2
🪙🪙 Tossing 2 Coins
S = {HH, HT, TH, TT}
Total outcomes = 4
S = {HH, HT, TH, TT}
Total outcomes = 4
🎲 Rolling 1 Die
S = {1, 2, 3, 4, 5, 6}
Total outcomes = 6
S = {1, 2, 3, 4, 5, 6}
Total outcomes = 6
🎲🎲 Rolling 2 Dice
S = {(1,1),(1,2),…,(6,6)}
Total outcomes = 36
S = {(1,1),(1,2),…,(6,6)}
Total outcomes = 36
Empirical (Experimental) Probability
Empirical probability is calculated by actually performing an experiment many times and observing the results. It is also called experimental probability.
P(E) = Number of times event E occurred / Total number of trials
Famous Coin-Toss Experiments
Mathematicians actually tossed coins thousands of times to study probability! Here are real results:
Mathematicians actually tossed coins thousands of times to study probability! Here are real results:
| Mathematician | Number of Tosses | Heads | Freq. of Heads |
|---|---|---|---|
| John Kerrich | 10,000 | 5,067 | 0.5067 |
| Karl Pearson | 24,000 | 12,012 | 0.5005 |
| Georges-Louis Buffon | 4,040 | 2,048 | 0.5069 |
Key Observation
As the number of trials increases, the empirical probability gets closer and closer to the theoretical probability. This is called the Law of Large Numbers. In the coin experiments above, all values approach 0.5 as trials increase!
As the number of trials increases, the empirical probability gets closer and closer to the theoretical probability. This is called the Law of Large Numbers. In the coin experiments above, all values approach 0.5 as trials increase!
📊 Empirical Probability — Step by Step
- Perform the experiment a large number of times (say n times).
- Count how many times the event E occurred — call this f (frequency of E).
- Calculate: P(E) = f / n
- Note: As n → ∞, empirical P(E) approaches theoretical P(E).
When is Empirical Probability Used?
When we cannot calculate exact chances mathematically — for example, finding the probability that a bus arrives on time, that a person is left-handed, or that a manufactured item is defective.
When we cannot calculate exact chances mathematically — for example, finding the probability that a bus arrives on time, that a person is left-handed, or that a manufactured item is defective.
✅ Worked Example
A bag of tomatoes is inspected. Out of 1000 tomatoes, 40 are found to be rotten. What is the probability of picking a rotten tomato?
P(rotten) = Number of rotten tomatoes / Total tomatoes = 40 / 1000 = 0.04 or 4/100
∴ P(good tomato) = 1 − 0.04 = 0.96
∴ P(good tomato) = 1 − 0.04 = 0.96
Theoretical (Classical) Probability
Theoretical probability is calculated without doing the experiment. It is based on reasoning and assumes all outcomes are equally likely.
P(E) = Number of outcomes favourable to E / Total number of possible outcomes (in Sample Space)
Important Condition!
Theoretical probability only works when all outcomes in the sample space are equally likely. For a fair coin or unbiased die, this condition holds.
Theoretical probability only works when all outcomes in the sample space are equally likely. For a fair coin or unbiased die, this condition holds.
📐 Range of Probability
- P(E) is always between 0 and 1 (including 0 and 1)
- P(impossible event) = 0
- P(sure/certain event) = 1
- Sum of probabilities of all outcomes in sample space = 1
📊 Visualising Probability on a Scale
Getting HEAD on a fair coin → P = 1/2 = 0.5
Rolling a 6 on a fair die → P = 1/6 ≈ 0.167
Drawing a red card from deck → P = 26/52 = 0.5
Drawing an Ace from deck → P = 4/52 ≈ 0.077
Probability with a Coin
One Coin Toss
When we toss a fair coin, there are 2 equally likely outcomes.
H
T
Sample Space S = {H, T} Total outcomes = 2P(Heads) = 1/2 = 0.5
P(Tails) = 1/2 = 0.5
P(H) + P(T) = 0.5 + 0.5 = 1 ✔ (sum of all probabilities)
P(Tails) = 1/2 = 0.5
P(H) + P(T) = 0.5 + 0.5 = 1 ✔ (sum of all probabilities)
Two Coins Tossed Together
When we toss 2 fair coins simultaneously, there are 4 equally likely outcomes.
H
H
HH
H
T
HT
T
H
TH
T
T
TT
Sample Space S = {HH, HT, TH, TT} Total outcomes = 4P(exactly 2 Heads) = 1/4 → {HH}
P(exactly 1 Head) = 2/4 = 1/2 → {HT, TH}
P(at least 1 Head) = 3/4 → {HH, HT, TH}
P(no Heads / 2 Tails) = 1/4 → {TT}
P(exactly 1 Head) = 2/4 = 1/2 → {HT, TH}
P(at least 1 Head) = 3/4 → {HH, HT, TH}
P(no Heads / 2 Tails) = 1/4 → {TT}
Common Mistake!
When tossing 2 coins, some students think there are only 3 outcomes: {2H, 1H1T, 2T}. This is WRONG because HT and TH are different outcomes. The correct sample space has 4 equally likely outcomes.
When tossing 2 coins, some students think there are only 3 outcomes: {2H, 1H1T, 2T}. This is WRONG because HT and TH are different outcomes. The correct sample space has 4 equally likely outcomes.
Three Coins Tossed Together
Sample Space S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Total outcomes = 8P(all 3 Heads) = 1/8
P(exactly 2 Heads) = 3/8 → {HHT, HTH, THH}
P(exactly 1 Head) = 3/8 → {HTT, THT, TTH}
P(at least 2 Heads) = 4/8 = 1/2
P(no Heads / 3 Tails)= 1/8
Total outcomes = 8P(all 3 Heads) = 1/8
P(exactly 2 Heads) = 3/8 → {HHT, HTH, THH}
P(exactly 1 Head) = 3/8 → {HTT, THT, TTH}
P(at least 2 Heads) = 4/8 = 1/2
P(no Heads / 3 Tails)= 1/8
Probability with a Die
Single Die
A standard die has 6 faces with numbers 1 to 6. Each face is equally likely.
1
2
3
4
5
6
Sample Space S = {1, 2, 3, 4, 5, 6} Total outcomes = 6P(getting 3) = 1/6
P(getting even number) = 3/6 = 1/2 → {2, 4, 6}
P(getting odd number) = 3/6 = 1/2 → {1, 3, 5}
P(getting > 4) = 2/6 = 1/3 → {5, 6}
P(getting prime) = 3/6 = 1/2 → {2, 3, 5}
P(getting ≤ 2) = 2/6 = 1/3 → {1, 2}
P(getting 7) = 0/6 = 0 (Impossible!)
P(getting ≤ 6) = 6/6 = 1 (Certain!)
P(getting even number) = 3/6 = 1/2 → {2, 4, 6}
P(getting odd number) = 3/6 = 1/2 → {1, 3, 5}
P(getting > 4) = 2/6 = 1/3 → {5, 6}
P(getting prime) = 3/6 = 1/2 → {2, 3, 5}
P(getting ≤ 2) = 2/6 = 1/3 → {1, 2}
P(getting 7) = 0/6 = 0 (Impossible!)
P(getting ≤ 6) = 6/6 = 1 (Certain!)
Two Dice Rolled Together
When 2 dice are rolled, the total number of outcomes = 6 × 6 = 36.
Two Dice Rolled Together
When 2 dice are rolled, the total number of outcomes = 6 × 6 = 36.
All 36 outcomes of rolling 2 dice. Purple cells show sum = 7 (6 ways — most likely sum!)
Total outcomes when 2 dice are rolled = 36P(sum = 7) = 6/36 = 1/6 ← Most likely sum!
P(sum = 2) = 1/36 → only (1,1)
P(sum = 12) = 1/36 → only (6,6)
P(sum = 6) = 5/36 → {(1,5),(2,4),(3,3),(4,2),(5,1)}
P(both same) = 6/36 = 1/6 → {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}
P(sum > 10) = 3/36 = 1/12 → {(5,6),(6,5),(6,6)}
P(sum = 2) = 1/36 → only (1,1)
P(sum = 12) = 1/36 → only (6,6)
P(sum = 6) = 5/36 → {(1,5),(2,4),(3,3),(4,2),(5,1)}
P(both same) = 6/36 = 1/6 → {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}
P(sum > 10) = 3/36 = 1/12 → {(5,6),(6,5),(6,6)}
Probability with a Deck of Playing Cards
A standard deck has 52 cards divided into 4 suits of 13 cards each.
Standard deck: 4 suits × 13 cards each = 52 cards total
Must Know: Deck of Cards Structure
Total cards = 52 | Suits = 4 (♠ Spades, ♣ Clubs, ♥ Hearts, ♦ Diamonds) | Cards per suit = 13 (A,2,3,4,5,6,7,8,9,10,J,Q,K) | Face cards (J,Q,K) = 12 | Aces = 4 | Black cards = 26 (♠+♣) | Red cards = 26 (♥+♦)
Total cards = 52 | Suits = 4 (♠ Spades, ♣ Clubs, ♥ Hearts, ♦ Diamonds) | Cards per suit = 13 (A,2,3,4,5,6,7,8,9,10,J,Q,K) | Face cards (J,Q,K) = 12 | Aces = 4 | Black cards = 26 (♠+♣) | Red cards = 26 (♥+♦)
One card is drawn at random from a well-shuffled deck of 52 cards.
Total outcomes = 52P(Ace) = 4/52 = 1/13
P(King) = 4/52 = 1/13
P(Face card) = 12/52 = 3/13 → J,Q,K of all 4 suits
P(Red card) = 26/52 = 1/2
P(Black card) = 26/52 = 1/2
P(Heart) = 13/52 = 1/4
P(Ace of Spades) = 1/52
P(Red King) = 2/52 = 1/26 → K♥ and K♦
P(not a Face card) = 40/52 = 10/13
P(a number card 2–10) = 36/52 = 9/13
Total outcomes = 52P(Ace) = 4/52 = 1/13
P(King) = 4/52 = 1/13
P(Face card) = 12/52 = 3/13 → J,Q,K of all 4 suits
P(Red card) = 26/52 = 1/2
P(Black card) = 26/52 = 1/2
P(Heart) = 13/52 = 1/4
P(Ace of Spades) = 1/52
P(Red King) = 2/52 = 1/26 → K♥ and K♦
P(not a Face card) = 40/52 = 10/13
P(a number card 2–10) = 36/52 = 9/13
Complementary Events
The complement of event E (written as Ē or E’) is the event “E does NOT happen.” Together, E and Ē cover all possibilities.
P(E) + P(Ē) = 1 ⟹ P(Ē) = 1 − P(E)
E and its complement Ē together cover the entire sample space
🎲 Die Example
E = getting a prime = {2,3,5}
P(E) = 3/6 = 1/2
P(Ē) = 1 − 1/2 = 1/2
Ē = {1, 4, 6} ✔
E = getting a prime = {2,3,5}
P(E) = 3/6 = 1/2
P(Ē) = 1 − 1/2 = 1/2
Ē = {1, 4, 6} ✔
🃏 Card Example
E = drawing a Face card
P(E) = 12/52 = 3/13
P(Ē) = 1 − 3/13 = 10/13
Ē = drawing a non-face card ✔
E = drawing a Face card
P(E) = 12/52 = 3/13
P(Ē) = 1 − 3/13 = 10/13
Ē = drawing a non-face card ✔
Smart Shortcut for Exams!
If finding P(E) directly is hard, find P(Ē) first (which may be easier), then use:
P(E) = 1 − P(Ē)
Example: P(at least one Head in 3 tosses) = 1 − P(no Heads) = 1 − 1/8 = 7/8
If finding P(E) directly is hard, find P(Ē) first (which may be easier), then use:
P(E) = 1 − P(Ē)
Example: P(at least one Head in 3 tosses) = 1 − P(no Heads) = 1 − 1/8 = 7/8
Impossible & Sure Events
🚫 Impossible Event
An event that can never happen.
Probability = 0Examples:
• Getting 7 on a die (faces: 1–6)
• Tossing a coin and getting both H and T simultaneously
• Drawing a green card from a standard deck
An event that can never happen.
Probability = 0Examples:
• Getting 7 on a die (faces: 1–6)
• Tossing a coin and getting both H and T simultaneously
• Drawing a green card from a standard deck
✅ Sure / Certain Event
An event that always happens.
Probability = 1Examples:
• Getting a number ≤ 6 on a die
• Getting H or T when tossing a coin
• Drawing a card from 52 is red or black
An event that always happens.
Probability = 1Examples:
• Getting a number ≤ 6 on a die
• Getting H or T when tossing a coin
• Drawing a card from 52 is red or black
Summary of Probability Range
Impossible (P=0) ← 0 ≤ P(E) ≤ 1 → Certain (P=1)
The closer P(E) is to 1, the more likely the event. The closer to 0, the less likely.
Impossible (P=0) ← 0 ≤ P(E) ≤ 1 → Certain (P=1)
The closer P(E) is to 1, the more likely the event. The closer to 0, the less likely.
📊 Comparing Likeliness of Events
| Event | P(E) | How Likely? |
|---|---|---|
| Getting 7 on a single die | 0 | Impossible ❌ |
| Getting tail on coin | 1/2 = 0.5 | Equally likely 🔄 |
| Rolling a number ≤ 4 | 4/6 = 0.67 | More likely ✅ |
| Drawing a non-Ace | 48/52 = 0.92 | Very likely ✅✅ |
| Drawing any card | 52/52 = 1 | Certain ✔✔✔ |
Chapter Summary — All Key Concepts
Definition of Probability
Measure of how likely an event is.
Always: 0 ≤ P(E) ≤ 1
Measure of how likely an event is.
Always: 0 ≤ P(E) ≤ 1
Empirical Probability
Based on experiment:
P(E) = freq(E) / total trials
Based on experiment:
P(E) = freq(E) / total trials
Theoretical Probability
Based on reasoning:
P(E) = favourable / total outcomes
Based on reasoning:
P(E) = favourable / total outcomes
Complementary Rule
P(E) + P(Ē) = 1
P(Ē) = 1 − P(E)
P(E) + P(Ē) = 1
P(Ē) = 1 − P(E)
Impossible Event
P = 0 (never happens)
e.g., rolling 7 on a die
P = 0 (never happens)
e.g., rolling 7 on a die
Sure / Certain Event
P = 1 (always happens)
e.g., rolling ≤ 6 on a die
P = 1 (always happens)
e.g., rolling ≤ 6 on a die
1 Coin: n = 2
S = {H, T}
P(H) = P(T) = 1/2
S = {H, T}
P(H) = P(T) = 1/2
2 Coins: n = 4
S = {HH, HT, TH, TT}
P(1 Head) = 2/4 = 1/2
S = {HH, HT, TH, TT}
P(1 Head) = 2/4 = 1/2
1 Die: n = 6
S = {1,2,3,4,5,6}
P(prime) = 3/6 = 1/2
S = {1,2,3,4,5,6}
P(prime) = 3/6 = 1/2
2 Dice: n = 36
P(sum=7) = 6/36 = 1/6
P(sum=2) = 1/36
P(sum=7) = 6/36 = 1/6
P(sum=2) = 1/36
Cards: n = 52
P(Ace) = 4/52 = 1/13
P(Heart) = 13/52 = 1/4
P(Ace) = 4/52 = 1/13
P(Heart) = 13/52 = 1/4
Sum Rule
Sum of P of all outcomes = 1
P(E₁)+P(E₂)+…+P(Eₙ) = 1
Sum of P of all outcomes = 1
P(E₁)+P(E₂)+…+P(Eₙ) = 1
🎲
Why is 7 the Most Common Sum with Two Dice?
7 can be made in 6 ways: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1). No other sum has this many combinations. That’s why 7 is the most important number in many dice games!
7 can be made in 6 ways: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1). No other sum has this many combinations. That’s why 7 is the most important number in many dice games!
🃏
Did You Know?
In a standard deck, there are exactly 4 Aces, 4 Kings, 4 Queens, and 4 Jacks (Jacks are also called Knaves). The Ace can be the highest or lowest card depending on the game!
In a standard deck, there are exactly 4 Aces, 4 Kings, 4 Queens, and 4 Jacks (Jacks are also called Knaves). The Ace can be the highest or lowest card depending on the game!
Important Exam Questions with Solutions
Q1. A coin is tossed 500 times. Heads appeared 280 times. Find the experimental probability of getting (i) Heads (ii) Tails.
(i) P(Heads) = 280/500 = 14/25 = 0.56
(ii) P(Tails) = (500−280)/500 = 220/500 = 11/25 = 0.44
Check: 14/25 + 11/25 = 25/25 = 1 ✔
(ii) P(Tails) = (500−280)/500 = 220/500 = 11/25 = 0.44
Check: 14/25 + 11/25 = 25/25 = 1 ✔
Q2. A die is thrown once. Find the probability of getting: (i) a prime number (ii) a number lying between 2 and 6 (iii) an odd number.
Total outcomes = 6
(i) Prime numbers = {2, 3, 5} → P = 3/6 = 1/2
(ii) Between 2 and 6 (exclusive) = {3, 4, 5} → P = 3/6 = 1/2
(iii) Odd = {1, 3, 5} → P = 3/6 = 1/2
(i) Prime numbers = {2, 3, 5} → P = 3/6 = 1/2
(ii) Between 2 and 6 (exclusive) = {3, 4, 5} → P = 3/6 = 1/2
(iii) Odd = {1, 3, 5} → P = 3/6 = 1/2
Q3. A card is drawn from a well-shuffled deck of 52 cards. Find the probability of drawing: (i) a Queen (ii) a red card (iii) a black King (iv) neither a Jack nor a King.
Total = 52
(i) Queens = 4 → P = 4/52 = 1/13
(ii) Red cards = 26 → P = 26/52 = 1/2
(iii) Black Kings = 2 (K♠, K♣) → P = 2/52 = 1/26
(iv) Jacks = 4, Kings = 4, total = 8. Neither J nor K = 52−8 = 44 → P = 44/52 = 11/13
(i) Queens = 4 → P = 4/52 = 1/13
(ii) Red cards = 26 → P = 26/52 = 1/2
(iii) Black Kings = 2 (K♠, K♣) → P = 2/52 = 1/26
(iv) Jacks = 4, Kings = 4, total = 8. Neither J nor K = 52−8 = 44 → P = 44/52 = 11/13
Q4. Two dice are thrown simultaneously. Find the probability that: (i) sum is 8 (ii) product is 12 (iii) both show same number.
Total outcomes = 36
(i) Sum=8: (2,6),(3,5),(4,4),(5,3),(6,2) = 5 ways → P = 5/36
(ii) Product=12: (2,6),(3,4),(4,3),(6,2) = 4 ways → P = 4/36 = 1/9
(iii) Same: (1,1),(2,2),(3,3),(4,4),(5,5),(6,6) = 6 ways → P = 6/36 = 1/6
(i) Sum=8: (2,6),(3,5),(4,4),(5,3),(6,2) = 5 ways → P = 5/36
(ii) Product=12: (2,6),(3,4),(4,3),(6,2) = 4 ways → P = 4/36 = 1/9
(iii) Same: (1,1),(2,2),(3,3),(4,4),(5,5),(6,6) = 6 ways → P = 6/36 = 1/6
Q5. In a class of 30 students, 6 are class monitors. If one student is selected at random, find: (i) P(selected is a monitor) (ii) P(selected is NOT a monitor).
Total students = 30
(i) P(monitor) = 6/30 = 1/5
(ii) P(not a monitor) = 1 − 1/5 = 4/5
OR: P(not monitor) = 24/30 = 4/5 ✔
(i) P(monitor) = 6/30 = 1/5
(ii) P(not a monitor) = 1 − 1/5 = 4/5
OR: P(not monitor) = 24/30 = 4/5 ✔
Q6. Three coins are tossed simultaneously. Find the probability of getting: (i) exactly 2 heads (ii) at least 2 heads (iii) at most 1 head.
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}, Total = 8
(i) Exactly 2 Heads: {HHT,HTH,THH} = 3 → P = 3/8
(ii) At least 2 Heads: {HHH,HHT,HTH,THH} = 4 → P = 4/8 = 1/2
(iii) At most 1 Head: {HTT,THT,TTH,TTT} = 4 → P = 4/8 = 1/2
(i) Exactly 2 Heads: {HHT,HTH,THH} = 3 → P = 3/8
(ii) At least 2 Heads: {HHH,HHT,HTH,THH} = 4 → P = 4/8 = 1/2
(iii) At most 1 Head: {HTT,THT,TTH,TTT} = 4 → P = 4/8 = 1/2
Q7. The probability that it rains on a particular day is 0.4. What is the probability that it does NOT rain?
P(rain) = 0.4
P(no rain) = 1 − P(rain) = 1 − 0.4 = 0.6
P(no rain) = 1 − P(rain) = 1 − 0.4 = 0.6
Q8. [HOTS] A bag contains 3 red, 5 blue, and 2 green balls. A ball is drawn at random. Find: (i) P(red) (ii) P(not green) (iii) P(blue or green).
Total balls = 3+5+2 = 10
(i) P(red) = 3/10
(ii) P(not green) = 1 − P(green) = 1 − 2/10 = 8/10 = 4/5
(iii) P(blue or green) = (5+2)/10 = 7/10
Note: P(red)+P(blue)+P(green) = 3/10+5/10+2/10 = 10/10 = 1
(i) P(red) = 3/10
(ii) P(not green) = 1 − P(green) = 1 − 2/10 = 8/10 = 4/5
(iii) P(blue or green) = (5+2)/10 = 7/10
Note: P(red)+P(blue)+P(green) = 3/10+5/10+2/10 = 10/10 = 1
Leave a Reply