Class 10 Mathematics – Probability | Interactive Visual Revision Notes

📝 Section 1 — Classroom Notes

🌎 Real-World Connection: A weather forecaster says “70% chance of rain tomorrow.” A doctor says “this medicine works for 9 out of 10 patients.” An insurance company calculates how likely an accident is before setting your premium. All of these rely on probability — the science of measuring chance — and this chapter gives you the tools to calculate exactly how likely any event is.

🎲 Topic 1: What Is Probability?

Probability measures the chance of an event.
It tells us how likely something is to happen.
We use it in daily life all the time.
Weather forecasts use probability.
It is always a number from 0 to 1.

🎲 Topic 2: Experiment and Outcome

An experiment is an action with results.
Each result is called an outcome.
Tossing a coin gives Head or Tail.
Rolling a die gives 1, 2, 3, 4, 5, or 6.
Picking a card from a deck is an experiment.
Outcomes must be equally likely for the formula.

🎲 Topic 3: Sample Space

The sample space is the set of all outcomes.
We write it as S.
For a coin: S = {Head, Tail}.
For a die: S = {1, 2, 3, 4, 5, 6}.
Total outcomes = number of items in S.
For two coins: S has 4 outcomes.
For two dice: S has 36 outcomes.

🎲 Topic 4: What Is an Event?

An event is one or more outcomes we want.
We write it as E.
Getting a 6 on a die is an event.
Getting an even number is also an event.
An event is a subset of the sample space.
One event can have many outcomes inside it.

🎲 Topic 5: Favorable Outcomes

Favorable outcomes are the ones we want.
They match the condition of our event.
For "getting even on a die": 2, 4, 6.
So favorable outcomes = 3.
We count only the outcomes that satisfy our event.
More favorable outcomes = higher probability.

🎲 Topic 6: Probability Formula

The formula is simple and powerful.
P(E) = favorable outcomes ÷ total outcomes.
P(even on die) = 3 ÷ 6 = 1/2.
All outcomes must be equally likely.
The answer is always a fraction or decimal.

P(E) = Number of favorable outcomes ÷ Total number of outcomes

🎲 Topic 7: Sure Event

A sure event always happens.
Its probability is 1.
Every outcome is favorable for it.
Example: getting less than 7 on a die.
All six faces are less than 7.
P(less than 7) = 6/6 = 1.

🎲 Topic 8: Impossible Event

An impossible event can never happen.
Its probability is 0.
No outcome is favorable for it.
Example: getting 8 on a regular die.
No face shows 8, so it is impossible.
P(getting 8) = 0/6 = 0.

🎲 Topic 9: Complementary Events

Two events are complementary if they cover all outcomes.
If E happens, then "not E" does not happen.
P(E) + P(not E) = 1
So P(not E) = 1 − P(E).
Example: P(Head) = 1/2.
P(not Head) = 1 − 1/2 = 1/2.
Very useful for hard "not" questions!

P(E) + P(not E) = 1

🎲 Topic 10: Key Rules of Probability

Probability is always between 0 and 1.
0 ≤ P(E) ≤ 1 for every event E.
The sum of all probabilities = 1.
P = 0 means impossible.
P = 1 means certain.
P closer to 1 = more likely.
P closer to 0 = less likely.

🎨 Section 2 — Easy Illustrations

Picture 1: Probability Scale

Probability always falls between 0 (impossible) and 1 (certain).

Picture 2: Sample Space of Two Dice

Two dice give 6 × 6 = 36 equally likely outcomes. Each cell is one pair.

Picture 3: Deck of Cards Overview

52 cards = 4 suits × 13 each. Red = Hearts + Diamonds. Black = Spades + Clubs.

Picture 4: Complementary Events

An event and its complement together cover all outcomes. Their probabilities add to 1.

Picture 5: Spinner Wheel of Chance

The spinner has 8 equal sections. Each color has 2 sections, so each probability is 1/4.

✏️ Section 3 — Worked Examples

Example 1: Balls in a Bag

Question: A bag has 5 red, 3 blue, and 2 green balls. One ball is drawn at random. Find P(red), P(blue), and P(not green).

1 Count total balls in the bag.

2 Total = 5 + 3 + 2 = 10 balls.

3 P(red) = red balls ÷ total = 5/10 = 1/2.

4 P(blue) = blue balls ÷ total = 3/10.

5 Green balls = 2, so P(green) = 2/10.

6 P(not green) = 1 − P(green) = 1 − 2/10 = 8/10 = 4/5.

P(red) = 1/2, P(blue) = 3/10, P(not green) = 4/5

Use complementary rule: P(not E) = 1 − P(E)

Example 2: Two Dice Thrown Together

Question: Two dice are thrown together. Find the probability that the sum of numbers is 9.

1 Total outcomes for two dice = 6 × 6 = 36.

2 List pairs that add up to 9.

3 (3, 6), (4, 5), (5, 4), (6, 3).

4 Favorable outcomes = 4.

5 P(sum = 9) = 4/36 = 1/9.

P(sum of 9) = 4/36 = 1/9

Always list the pairs carefully. Order matters!

Example 3: Drawing a Card from a Deck

Question: A card is drawn from a well-shuffled deck of 52 cards. Find P(queen), P(red card), and P(face card).

1 Total cards in a deck = 52.

2 Queens in a deck = 4 (one per suit).

3 P(queen) = 4/52 = 1/13.

4 Red cards = 26 (13 Hearts + 13 Diamonds).

5 P(red card) = 26/52 = 1/2.

6 Face cards = 12 (4 Jacks + 4 Queens + 4 Kings).

7 P(face card) = 12/52 = 3/13.

P(queen) = 1/13, P(red) = 1/2, P(face card) = 3/13

Remember: 4 suits, 13 cards each, 12 face cards total

📊 Section 4 — Cheat Sheet

Term	What It Means	Key Fact
Probability	Chance of an event happening	0 ≤ P ≤ 1
Experiment	An action with outcomes	Toss, roll, pick
Outcome	One possible result	Head, Tail, 1–6
Sample Space	Set of all outcomes	S = {all results}
Event	Outcomes we are looking for	Subset of S
Favorable	Outcomes matching our event	Count them carefully
P(E) Formula	Favorable ÷ Total	Must be equally likely
Sure Event	Always happens	P = 1
Impossible Event	Never happens	P = 0
Complementary	P(E) + P(not E) = 1	Cover all outcomes
Two Dice	36 total outcomes	6 × 6 grid
Deck of Cards	52 cards, 4 suits	12 face, 26 red

📄 Section 5 — Take-Home Handout

📝 A. Fill in the Blanks

1. Probability of any event lies between _________ and _________.
2. The set of all possible outcomes is called the _________.
3. P(E) + P(not E) = _________.
4. If P(E) = 0, the event is called _________.
5. Two dice thrown together give _________ total outcomes.

✓ B. True or False

1. Probability can be negative. _______
2. A sure event has probability 1. _______
3. A deck of cards has 48 cards. _______
4. The sum of all probabilities of an experiment equals 1. _______
5. Getting a 7 on a single die is an impossible event. _______

✏️ C. Solve These

1. A box has 4 red, 6 white, and 5 black marbles. One marble is drawn. Find P(white) and P(not red).

Show your work here

2. Two dice are thrown together. Find the probability that the sum of numbers on top is 7.

Show your work here

🔗 D. Match the Columns

Column A	Column B
1. P(sure event)	a. 0
2. P(impossible event)	b. 1/2
3. P(Head on a coin)	c. 1
4. P(E) + P(not E)	d. 1

🎨 E. Draw and Solve

Draw a spinner with 6 equal sections: 3 red, 2 blue, 1 green. Find the probability of landing on each color.

Draw your spinner and write probabilities here

✅ Detailed Solutions

Solution to Q1: Probability from a Bag of Marbles

Question: A box has 4 red, 6 white, and 5 black marbles. One marble is drawn at random. Find P(white) and P(not red).

1 Total marbles = 4 + 6 + 5 = 15.

2 P(white) = favorable ÷ total = 6 ÷ 15 = 2/5.

3 P(red) = 4 ÷ 15.

4 P(not red) = 1 − P(red) = 1 − 4/15 = 11/15.

∴ P(white) = 2/5 | P(not red) = 11/15

Solution to Q2: Sum of 7 on Two Dice

Question: Two dice are thrown together. Find the probability that the sum of the numbers on top is 7.

1 Total outcomes = 6 × 6 = 36.

2 List all pairs (Die 1, Die 2) that add up to 7:

3 (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 favorable outcomes.

4 P(sum = 7) = 6 ÷ 36 = 1/6.

∴ P(sum = 7) = 6/36 = 1/6

🃏 Section 6 — Revision Flashcards

👉

🎲

What is the formula for probability?

P(E) = favorable outcomes ÷ total outcomes.

All outcomes must be equally likely.

📉

What is the range of probability?

Probability is always between 0 and 1 (inclusive).

0 ≤ P(E) ≤ 1

💡

What is a sample space?

The set of all possible outcomes of an experiment.

Die: {1,2,3,4,5,6} — Coin: {H,T}

✅

What is a sure event?

An event that always happens.

P(sure event) = 1

❌

What is an impossible event?

An event that can never happen.

P(impossible event) = 0

🔄

What are complementary events?

P(E) + P(not E) = 1

If you know one, subtract from 1 to find the other.

🃏

How many face cards in a standard deck?

12 face cards in total.

4 Jacks + 4 Queens + 4 Kings

🎲

How many outcomes when two dice are thrown?

36 outcomes in total.

6 × 6 = 36 (each die has 6 faces)

Class 10 Mathematics — Probability

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🎲 Class 10 Mathematics — Probability Interactive Visual Revision Notes Aligned with the Class 10 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.

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