Two events E and Ē are complementary if they are the only two possible outcomes and exactly one of them must occur. The key property is P(E) + P(Ē) = 1, which gives P(Ē) = 1 − P(E).

Q: How many outcomes are there when two dice are thrown?

When two dice are thrown simultaneously, the total number of equally likely outcomes is 6 × 6 = 36. Each outcome is an ordered pair (a, b) where a is the result of the first die and b of the second.

Q: How many face cards are in a deck of 52 cards?

A standard deck has 12 face cards: King, Queen, and Jack of each of the 4 suits (spades, hearts, diamonds, and clubs). The 4 suits give 3 × 4 = 12 face cards total.

Q: What is the probability of an impossible event and a sure event?

The probability of an impossible event is 0, for example, rolling a 7 on a standard 6-faced die. The probability of a sure (certain) event is 1, for example, rolling a number less than 7 on a standard die.

Chapter 14: Probability Notes for Class 10 gives you a clear, NCERT-aligned guide to the basic principles and calculations of probability commonly tested in CBSE exams. Learn definitions and notation for experiments, outcomes, sample space, and events, plus the formal definition of probability and methods to compute probabilities for simple, complementary, mutually exclusive, and independent events. It includes step-by-step worked examples, shortcut tips for quick computation, revision bullets, practice problems with answers, and common mistake alerts to build confidence and strong conceptual understanding for exam success.

What is Probability?
Key Terms of Probability
Types of Events in Probability
Tossing Coins Problem
Two Dice Problems
Understanding Playing Cards
Previous Year Board Questions

What is Probability?

Definition: The theoretical probability of an event E is:

P(E) = Number of outcomes favourable to E / Number of all possible outcomes

This formula assumes all outcomes are equally likely.

Experimental vs Theoretical Probability

Feature	Experimental Probability	Theoretical Probability
Based on	Actual trials or observations	Logical assumptions and reasoning
Formula	Favourable trials ÷ Total trials	Favourable outcomes ÷ Total outcomes
Requires	Repeated experiments	Equally likely outcomes
Also Called	Empirical Probability	Classical Probability

As the number of trials increases, experimental probability gets closer and closer to theoretical probability.

Key Terms of Probability

Term	Meaning	Example
Experiment	An action with well-defined outcomes.	Tossing a coin, throwing a die
Outcome	A possible result of one trial.	Getting Head (H) on a coin toss
Sample Space (S)	The set of all possible outcomes of an experiment.	(S = {H, T}) for a coin toss
Event (E)	A specific outcome or a set of outcomes of interest.	Getting an even number on a die
Elementary Event	An event that contains exactly one outcome.	Getting 3 on a die
Favourable Outcomes	Outcomes that satisfy the given event condition.	For the event ‘getting an even number’: {2, 4, 6}

Sum of Elementary Events

The sum of probabilities of all elementary events of an experiment always equals 1.

Example: For a die: P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1/6 × 6 = 1

Types of Events in Probability

Impossible Event: P(E) = 0

e.g., rolling 8 on a die

Sure/Certain Event: P(E) = 1

e.g., rolling <7 on a die

For any event E: 0 ≤ P(E) ≤ 1
Complementary Event (Not E): P(Ē) = 1 − P(E)

Complementary Events Explained

If event E is ‘getting a head’, then Ē (complement, read as ‘E-bar’) is ‘not getting a head’ = ‘getting a tail’. They are complementary because together the events cover all possibilities.

Complementary Events Formula

P(E) + P(Ē) = 1 ⟹ P(Ē) = 1 − P(E)

Tossing Coins Problem

For two coins, the possible outcomes are:

Coin 1	Coin 2	Outcome
H	H	HH
H	T	HT
T	H	TH
T	T	TT

Total Outcomes = 4

Therefore, when two coins are tossed together, the sample space is:

S = {HH,HT,TH,TT}

Examples:

1. Probability of getting two heads

Favourable outcome = {HH}

P(Two Heads) = 1/4

2. Probability of getting two tails

Favourable outcome = {TT}

P(Two Tails) = 1/4

3. Probability of getting one head and one tail

Favourable outcomes = {HT, TH}

P(One Head and One Tail) = 2/4 = ½

Number of outcomes = $%202%5En$

where n = number of coins.

Two Dice Problems

When two dice are thrown, the total number of outcomes = 6 × 6 = 36. Each ordered pair (a, b) represents a unique outcome where a ≠ b position-wise, so (1, 4) ≠ (4, 1).

two die

Event	Favourable Outcomes	Probability
Sum = 8	(2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes	\frac{5}{36}
Sum = 13	None (Impossible event)	0
Sum ≤ 12	All 36 possible outcomes (Sure event)	1
Sum = 7	(1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes	\frac{6}{36} = \frac{1}{6}
Sum = 2	(1,1) → 1 outcome	\frac{1}{36}
Sum = 12	(6,6) → 1 outcome	\frac{1}{36}

Understanding Playing Cards

A standard deck has 52 cards.

♠ Spades

Black - 13 cards

♥ Hearts

Red - 13 cards

♦ Diamonds

Red - 13 cards

♣ Clubs

Black - 13 cards

Card Type	Count	Details
Total Cards	52	4 suits × 13 cards each
Red Cards (♥ + ♦)	26	Hearts and Diamonds
Black Cards (♠ + ♣)	26	Spades and Clubs
Aces	4	One Ace in each suit
Face Cards (K, Q, J)	12	3 face cards × 4 suits
Number Cards (2–10)	36	9 number cards × 4 suits
Kings of Red Suits	2	King of Hearts and King of Diamonds

Previous Year Board Questions

Question 1: A bag contains 6 red, 4 white and 8 blue balls. A ball is drawn at random. Find the probability that the ball drawn is (i) blue, (ii) not red.

Solution: Total = 18

Blue balls = 8

Red balls = 6

White balls = 4

(i) P(blue) = 8/18 = 4/9

(ii) P(not red) = 12/18 = 2/3

Question 2: Cards marked 2 to 101 are placed in a box and one is drawn at random. Find the probability that the number on the card is (i) a perfect square, (ii) divisible by 7.

Solution: Total = 100

(i) Perfect squares: 4, 9, 16, 25, 36, 49, 64, 81 and 100

There are 9 perfect squares

P(a perfect square) = 9/100

(ii) Divisible by 7 (7, 14, ... 98)

There are 14 numbers divisible by 7

P(divisible by 7) = 14/100 = 7/50

Question 3: A die is thrown twice. What is the probability that (i) 5 will not come up either time? (ii) 5 will come up at least once?

Solution: Total = 36

Outcomes where 5 appears: (5,1), (5,2)...(5,6), (1,5)...(6,5) minus double-count Therefore 11 outcomes

(i) P(5 not at all) = 25/36

(ii) P(5 at least once) = 11/36

Question 4: A number x is selected at random from the numbers 1, 2, 3 and 4. Another number y is selected at random from 1, 2, 3 and 4. Find the probability that the product of x and y is less than 9.

Solution: Total pairs = 16

Products ≥ 9: (3,3)=9, (3,4)=12, (4,3)=12, (4,4)=16

Total 4 favourable outcomes

P(xy < 9) = 12/16 = 3/4

Click below to download your free Class 10 Maths Chapter 14: Probability PDF Notes perfect for last-minute CBSE board exam revision.

Class 10 Maths Chapter 14: Probability PDF Notes

Frequently Asked Questions of Chapter 14: Probability Notes for Class 10

1. What are complementary events in probability?

Two events E and Ē are complementary if they are the only two possible outcomes and exactly one of them must occur. The key property is P(E) + P(Ē) = 1, which gives P(Ē) = 1 − P(E).

2. How many outcomes are there when two dice are thrown?

When two dice are thrown simultaneously, the total number of equally likely outcomes is 6 × 6 = 36. Each outcome is an ordered pair (a, b) where a is the result of the first die and b of the second.

3. How many face cards are in a deck of 52 cards?