Curriculum

Mathematics Grade 10

Grade 10 Mathematics

0/14

Interactive Worksheets

0/0

Chapter Wise Worksheets

0/14

Solved Problems and solutions

0/2

Text lesson

Chapter 14: PROBABILITY

Grade 10 Mathematics · Chapter 14

Probability

Understand the mathematics of chance — from theoretical probability and complementary events to dice, cards, coins and real-world applications. Build the foundation for confident exam success.

🎲 Dice & Coins 🃏 Playing Cards 🎯 Board Exam Ready

📋 Contents

1. Introduction & History
2. Key Definitions
3. Probability Formula
4. Types of Events

5. Complementary Events
6. Playing Cards Guide
7. Worked Examples (10+)
8. Practice Sets A–D

📖	Section 1 — Introduction & Brief History

Probability is the branch of mathematics that deals with measuring the likelihood of events. Every day we make judgements like “it will probably rain today” or “I will likely pass the exam” — probability gives these ideas a precise numerical value.

Brief History of Probability:
The formal study of probability began in the 16th century. An Italian scholar named Gerolamo Cardano wrote one of the earliest works on games of chance. Later, great mathematicians including James Bernoulli (1654–1705), Abraham de Moivre (1667–1754), and Pierre Simon Laplace (1749–1827) developed the subject into a rigorous mathematical theory. Laplace’s work in 1812 is considered the most significant single contribution to probability theory. Today, probability is used in medicine, finance, engineering, genetics, artificial intelligence, and many other fields.

Two Types of Probability

Type	How it works	Example
Empirical (Experimental)	Based on actual observations from repeated experiments	Tossing a coin 1000 times and counting heads
Theoretical (Classical)	Based on logical reasoning and assumptions about equally likely outcomes	P(head) = 1/2 because heads and tails are equally likely

In this chapter, we focus on theoretical probability and always assume that all outcomes are equally likely.

📚	Section 2 — Key Definitions You Must Know

Term	Definition & Example
Experiment	Any action with a well-defined set of results. Example: tossing a coin, rolling a die.
Outcome	A single result of an experiment. Example: getting “Head” when a coin is tossed.
Sample Space	The complete set of all possible outcomes. Example: {H, T} for a coin toss; {1,2,3,4,5,6} for a die.
Event	Any subset of the sample space — one or more outcomes we are interested in. Example: getting an even number when rolling a die = {2, 4, 6}.
Elementary Event	An event with exactly one outcome. Example: getting 3 when rolling a die.
Equally Likely Outcomes	Outcomes that have the same chance of happening. Example: each face of a fair die has an equal 1/6 chance.
Favourable Outcomes	Outcomes that satisfy the condition of the event. Example: for event “getting odd”, favourable outcomes are {1, 3, 5}.
Trial	One performance of the experiment. Each coin toss or die throw is one trial.

💡 Remember: The sum of probabilities of all elementary events of any experiment always equals 1. This is because one of the outcomes must certainly happen.

🔢	Section 3 — The Probability Formula

The theoretical (classical) probability of an event E is defined as:

P(E) = Number of Favourable Outcomes ÷ Total Number of Possible Outcomes

This formula applies when all outcomes are equally likely.

Important Properties of Probability

No.	Property	Meaning
1	0 ≤ P(E) ≤ 1	Probability is always between 0 and 1 (inclusive)
2	P(E) = 0	Impossible event — can never happen (e.g., rolling a 7 on a standard die)
3	P(E) = 1	Certain (sure) event — always happens (e.g., rolling a number less than 7)
4	Σ P(all elementary events) = 1	Probabilities of all elementary events in an experiment always sum to 1

Probability Scale (0 to 1)

Impossible
P = 0

Very
Unlikely

Unlikely

Even
Chance
P = 0.5

Likely

Very
Likely

Certain
P = 1

🏷️	Section 4 — Types of Events

Impossible Event An event that can never occur. No favourable outcomes exist. P = 0 Example: Rolling a 9 on a standard die.	Sure (Certain) Event An event that always occurs. All outcomes are favourable. P = 1 Example: Rolling a number less than 7 on a die.
Elementary Event An event containing exactly one outcome from the sample space. Example: Getting exactly 4 when rolling a die. Sum of all elementary event probabilities = 1.	Compound Event An event with two or more outcomes from the sample space. Example: Getting an even number {2, 4, 6} when rolling a die. P = 3/6 = 1/2

🔄	Section 5 — Complementary Events

For any event E, the complement of E (written as Ē or E’) is the event that E does not happen. Together, E and Ē cover every possible outcome.

P(E) + P(Ē) = 1
→ P(Ē) = 1 − P(E)

Complement Diagram

P(E) Event E occurs	P(Ē) = 1 − P(E) Event E does NOT occur
P(E) + P(Ē) = 1 (entire sample space)

Practical Use: If finding P(E) directly is complex, calculate 1 − P(Ē) instead. For example: P(at least one head) = 1 − P(no head) is often easier to compute.

🃏	Section 6 — A Deck of Playing Cards (Complete Guide)

A standard deck has 52 cards divided into 4 suits of 13 cards each. Knowing this structure is essential for probability questions in exams.

Suit	Symbol	Colour	Cards (13 total)
Spades	♠	Black	A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K
Clubs	♣	Black	A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K
Hearts	♥	Red	A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K
Diamonds	♦	Red	A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K

Quick Reference: Card Counts

Category	Count	Which cards?
Aces	4	A of ♠, ♣, ♥, ♦
Kings	4	K of ♠, ♣, ♥, ♦
Queens	4	Q of ♠, ♣, ♥, ♦
Jacks	4	J of ♠, ♣, ♥, ♦
Face Cards (J, Q, K)	12	3 face cards × 4 suits
Red cards	26	All Hearts + all Diamonds
Black cards	26	All Spades + all Clubs
Red face cards	6	J, Q, K of Hearts + J, Q, K of Diamonds
Number cards (2–10)	36	9 number cards × 4 suits

✏️	Section 7 — Worked Examples (12 Solved Problems)

EXAMPLE 1 Coin Toss — Basic Probability

Question: A fair coin is tossed once. Find the probability of (i) getting a Head (ii) getting a Tail.

👁 Show Solution

Sample space = {H, T}. Total possible outcomes = 2.

(i) Favourable outcomes for Head = 1. P(Head) = 1/2 = 0.5

(ii) Favourable outcomes for Tail = 1. P(Tail) = 1/2 = 0.5

Note: P(Head) + P(Tail) = 1/2 + 1/2 = 1 ✓ (sum of all elementary events)

EXAMPLE 2 Single Die — Multiple Events

Question: A fair die is thrown once. Find the probability of: (i) getting a 6, (ii) getting a prime number, (iii) getting a number greater than 4, (iv) getting a number less than 7.

👁 Show Solution

Sample space = {1, 2, 3, 4, 5, 6}. Total outcomes = 6.

(i) Favourable for 6 = {6} → P(6) = 1/6

(ii) Prime numbers ≤ 6: {2, 3, 5} → P(prime) = 3/6 = 1/2

(iii) Greater than 4: {5, 6} → P(>4) = 2/6 = 1/3

(iv) Less than 7: {1,2,3,4,5,6} → all outcomes! P(<7) = 6/6 = 1 (certain event)

EXAMPLE 3 Complementary Events

Question: In a bag of 20 marbles, 7 are red and the rest are blue. One marble is picked at random. Find: (i) P(red marble), (ii) P(not red).

👁 Show Solution

Total marbles = 20. Red = 7. Blue = 20 − 7 = 13.

(i) P(red) = 7/20

(ii) Method 1: P(not red) = 13/20

Method 2 (using complement): P(not red) = 1 − P(red) = 1 − 7/20 = 13/20

EXAMPLE 4 Playing Cards

Question: A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability that it is: (i) a King, (ii) a red card, (iii) a face card, (iv) not a face card.

👁 Show Solution

Total cards = 52.

(i) Kings = 4. P(King) = 4/52 = 1/13

(ii) Red cards = 26. P(red) = 26/52 = 1/2

(iii) Face cards (J, Q, K × 4 suits) = 12. P(face card) = 12/52 = 3/13

(iv) P(not face card) = 1 − 3/13 = 10/13

EXAMPLE 5 Two Coins Tossed Simultaneously

Question: Two fair coins are tossed together. Find the probability of: (i) getting exactly one head, (ii) getting at least one head, (iii) getting no head.

👁 Show Solution

Sample space = {HH, HT, TH, TT}. Total outcomes = 4.

(i) Exactly one head: {HT, TH} → 2 outcomes. P = 2/4 = 1/2

(ii) At least one head: {HH, HT, TH} → 3 outcomes. P = 3/4

(iii) No head: {TT} → 1 outcome. P = 1/4

Verification: 1/2 is for exactly one; 1/4 for two heads; 1/4 for no heads → total = 1 ✓

EXAMPLE 6 Two Dice Thrown — Sample Space of 36

Question: Two dice are thrown at the same time. Find the probability that the sum of the numbers on top is: (i) 7, (ii) 11, (iii) less than 5.

Sample Space Grid (Die 1 vs Die 2):

D1\D2	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)

Green = sum 7 | Yellow = sum <5

👁 Show Solution

Total possible outcomes = 6 × 6 = 36

(i) Sum = 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → 6 pairs. P = 6/36 = 1/6

(ii) Sum = 11: (5,6),(6,5) → 2 pairs. P = 2/36 = 1/18

(iii) Sum < 5: sums of 2, 3, 4 → (1,1),(1,2),(2,1),(1,3),(3,1),(2,2) → 6 pairs. P = 6/36 = 1/6

EXAMPLE 7 Balls in a Bag — Mixed Colours

Question: A bag contains 5 red, 4 blue, and 3 green balls. One ball is drawn at random. Find: (i) P(red), (ii) P(blue or green), (iii) P(not red).

👁 Show Solution

Total balls = 5 + 4 + 3 = 12.

(i) P(red) = 5/12

(ii) Blue or green = 4 + 3 = 7. P(blue or green) = 7/12

(iii) P(not red) = 1 − 5/12 = 7/12 ✓ (same as blue or green)

EXAMPLE 8 Numbers on Discs / Numbered Cards

Question: Cards numbered 1 to 30 are placed in a box. One card is drawn at random. Find the probability that it bears: (i) a multiple of 5, (ii) a perfect square, (iii) a prime number.

👁 Show Solution

Total cards = 30.

(i) Multiples of 5 up to 30: {5,10,15,20,25,30} → 6 cards. P = 6/30 = 1/5

(ii) Perfect squares up to 30: {1,4,9,16,25} → 5 cards. P = 5/30 = 1/6

(iii) Primes up to 30: {2,3,5,7,11,13,17,19,23,29} → 10 cards. P = 10/30 = 1/3

EXAMPLE 9 Three Coins Tossed — Finding All Outcomes

Question: Three coins are tossed simultaneously. Find the probability of getting: (i) exactly two tails, (ii) at least two heads, (iii) all heads.

👁 Show Solution

Sample space = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Total = 8.

(i) Exactly two tails: {HTT, THT, TTH} → 3 outcomes. P = 3/8

(ii) At least two heads (2 or 3 heads): {HHH, HHT, HTH, THH} → 4 outcomes. P = 4/8 = 1/2

(iii) All heads: {HHH} → 1 outcome. P = 1/8

EXAMPLE 10 Defective Items — Real World Application

Question: A box contains 200 pens of which 30 are defective. One pen is chosen at random. Find: (i) P(pen is defective), (ii) P(pen works fine).

👁 Show Solution

Total pens = 200. Defective = 30. Working = 170.

(i) P(defective) = 30/200 = 3/20 = 0.15

(ii) P(working) = 170/200 = 17/20 = 0.85

Check: 0.15 + 0.85 = 1.00 ✓

EXAMPLE 11 Letters on Faces of a Die

Question: A special die has its six faces labelled: P, R, O, B, A, B. The die is thrown once. Find: (i) P(B), (ii) P(vowel).

👁 Show Solution

Faces: {P, R, O, B, A, B}. Total = 6.

(i) Letter B appears twice. P(B) = 2/6 = 1/3

(ii) Vowels in {P,R,O,B,A,B}: O and A → 2 faces. P(vowel) = 2/6 = 1/3

EXAMPLE 12 Piggy Bank / Mixed Coins Problem

Question: A piggy bank has: 80 coins worth ₹1, 50 coins worth ₹2, 30 coins worth ₹5, and 40 coins worth ₹10. One coin falls out randomly. Find: (i) P(₹5 coin), (ii) P(not ₹10 coin).

👁 Show Solution

Total coins = 80 + 50 + 30 + 40 = 200.

(i) ₹5 coins = 30. P(₹5) = 30/200 = 3/20

(ii) ₹10 coins = 40. P(₹10) = 40/200 = 1/5.

P(not ₹10) = 1 − 1/5 = 4/5

📝	Section 8 — Practice Sets A, B, C & D

Practice Set A — Basic Probability

1. A bag has 4 red, 6 blue, and 2 yellow balls. One ball is drawn at random. Find: (a) P(yellow), (b) P(not blue), (c) P(red or yellow).

2. A fair die is thrown. Find the probability of getting: (a) a factor of 6, (b) a number divisible by 3, (c) a number that is both prime and odd.

3. If P(E) = 0.35, find P(Ē). What type of number must any probability value always be?

👁 Answers

1. Total = 12. (a) P(yellow) = 2/12 = 1/6 (b) P(not blue) = 6/12 = 1/2 (c) P(red or yellow) = 6/12 = 1/2

2. (a) Factors of 6: {1,2,3,6} → 4/6 = 2/3 (b) Div by 3: {3,6} → 2/6 = 1/3 (c) Prime and odd: {3,5} → 2/6 = 1/3

3. P(Ē) = 1 − 0.35 = 0.65. Probability must always be a number between 0 and 1 inclusive.

Practice Set B — Coins and Dice

1. Two dice are thrown. Find the probability that: (a) both show the same number, (b) the product of the two numbers is 12, (c) the sum is a prime number.

2. A coin is tossed three times. What is the probability of getting exactly two heads?

3. A die is thrown twice. Find the probability that 4 appears at least once.

👁 Answers

1. Total = 36. (a) Same: {(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)} → 6/36 = 1/6 (b) Product 12: (2,6)(3,4)(4,3)(6,2) → 4/36 = 1/9 (c) Prime sums (2,3,5,7,11): count pairs → 15 outcomes → 15/36 = 5/12

2. Sample space = 8. Exactly two heads: HHT, HTH, THH → 3/8

3. P(4 not appearing at all) = (5/6)² = 25/36. P(at least one 4) = 1 − 25/36 = 11/36

Practice Set C — Playing Cards

1. From a deck of 52 cards, one card is drawn at random. Find: (a) P(an ace of black suit), (b) P(a red king), (c) P(jack of diamonds).

2. Five cards — A, 2, 3, 4, 5 of Hearts — are shuffled. One is picked. Then without replacing it, another is picked. What is the probability the second card is the Ace if the first was the 5?

3. A card is drawn from 52 cards. Find: (a) P(neither a heart nor a king), (b) P(a king or a queen).

👁 Answers

1. (a) Black aces (Ace of Spades + Ace of Clubs) = 2 → 2/52 = 1/26 (b) Red kings = 2 → 2/52 = 1/26 (c) Jack of diamonds = 1 → 1/52

2. After 5 is removed, 4 cards remain: A,2,3,4. P(Ace next) = 1/4

3. (a) Hearts = 13, Kings = 4, King of Hearts counted once → Hearts or kings = 13+3 = 16. Neither = 52−16 = 36 → 36/52 = 9/13 (b) Kings + Queens = 8 → 8/52 = 2/13

Practice Set D — Mixed & Word Problems

1. In a class of 36 students, 20 like cricket, 12 like football, and 4 like both. A student is chosen randomly. Find P(the student likes cricket but not football).

2. The probability that it rains on a given day is 0.4. What is the probability that it does NOT rain on that day?

3. A box has 90 numbered discs (1 to 90). Find the probability of drawing: (a) a two-digit number, (b) a multiple of both 3 and 5, (c) a perfect square.

👁 Answers

1. Cricket only (not football) = 20 − 4 = 16. P = 16/36 = 4/9

2. P(no rain) = 1 − 0.4 = 0.6

3. Total = 90. (a) Two-digit: 10 to 90 = 81 numbers → 81/90 = 9/10 (b) Multiple of 15: {15,30,45,60,75,90} = 6 → 6/90 = 1/15 (c) Perfect squares: {1,4,9,16,25,36,49,64,81} = 9 → 9/90 = 1/10

📚	Chapter Summary — Probability at a Glance

🔢 Core Formula

P(E) = Favourable Outcomes ÷ Total Outcomes
Always: 0 ≤ P(E) ≤ 1
Impossible event: P = 0
Certain event: P = 1

🎲 Sample Space Sizes

1 coin → 2 outcomes
2 coins → 4 outcomes
3 coins → 8 outcomes
1 die → 6 outcomes
2 dice → 36 outcomes
1 card from deck → 52 outcomes

🔄 Complementary Events

P(E) + P(Ē) = 1
P(Ē) = 1 − P(E)

Use complement when direct counting is harder than counting what does NOT happen.

🃏 Key Card Facts

52 cards = 4 suits × 13 cards
Face cards = 12 (J, Q, K × 4)
Red = 26, Black = 26
Aces = 4, Kings = 4
Red face cards = 6

🎯	Exam Quick-Check — 8 Must-Know Points

1	Probability is always a number between 0 and 1. Negative probabilities and values greater than 1 are impossible.
2	The sum of probabilities of all elementary events in any experiment equals exactly 1.
3	Complementary rule: P(Ē) = 1 − P(E). Always use this when counting “not happening” is easier.
4	For two dice, always use a 6×6 = 36 outcome grid. Never assume 11 outcomes (the sums 2–12 are not equally likely!).
5	A deck has 52 cards: 4 suits × 13 cards, 12 face cards, 4 aces, 26 red, 26 black. Memorise these.
6	When a card is drawn without replacement, the total number of outcomes changes for the second draw.
7	“At least one” problems are often easiest solved as: P(at least one) = 1 − P(none).
8	Theoretical probability assumes equally likely outcomes — verify this condition is met before applying the formula.

⚠️ Common Mistakes to Avoid

✗	Writing P(E) as a value greater than 1 or less than 0 — this is always wrong.
✗	Assuming the 11 possible sums (2 to 12) of two dice are equally likely — they are NOT.
✗	Counting (H,T) and (T,H) as the same outcome when two coins are tossed — they are different outcomes.
✗	Forgetting that face cards are only Jacks, Queens and Kings — Aces are NOT face cards.

About This Chapter: This page covers Grade 10 Probability — one of the most practical and frequently tested chapters in secondary mathematics. Students learn the concept of theoretical or classical probability, how to enumerate sample spaces, calculate probabilities using the fundamental formula, and apply the complementary event rule. Topics include probability of events involving coins, dice, playing cards, numbered cards, and real-life situations like defective items and mixed collections. The chapter also clarifies the difference between experimental (empirical) probability and theoretical probability, and explains why outcomes must be equally likely for the formula to apply. The impossible event (P=0) and certain event (P=1) are important boundary concepts. Mastery of probability in Class 10 builds the foundation for higher-level statistics and data science.