Probability

Board: CBSE | Class: Class 9

Comprehensive study notes for Probability by Ajay Yadav (Math King of Katargam). Master every concept with clear explanations, solved examples, and practice problems.

Key Concepts

Introduction

Probability measures the likelihood of an event occurring. It ranges from 0 (impossible) to 1 (certain). Experiment: A process with well-defined outcomes. Random experiment: Cannot predict the outcome with certainty.

Trial and Event

A trial is a single performance of an experiment. An event is a specific outcome or set of outcomes. Empirical probability: Based on actual experiments and observations.

Empirical Probability Formula

P(E) = (Number of times event E occurs) / (Total number of trials). For example, if a coin is tossed 100 times and heads appears 52 times, P(heads) = 52/100 = 0.52.

Key Properties

0 ≤ P(E) ≤ 1. P(E) + P(not E) = 1. If P(E) = 0, the event is impossible. If P(E) = 1, the event is certain. As the number of trials increases, the empirical probability approaches the theoretical probability.

Coin Toss

For a fair coin: P(Heads) = 1/2, P(Tails) = 1/2. For two coins: P(both heads) = 1/4, P(one head one tail) = 1/2, P(both tails) = 1/4.

Dice Roll

For a fair die: P(getting any number 1-6) = 1/6. P(even number) = 3/6 = 1/2. P(number > 4) = 2/6 = 1/3.

Cards

A standard deck has 52 cards: 4 suits (hearts, diamonds, clubs, spades), 13 cards each. Hearts & diamonds are red; clubs & spades are black. Face cards: Jack, Queen, King (12 total).

Important Formulas

Empirical Probability	P(E) = (Favorable outcomes)/(Total trials)
Complement	P(not E) = 1 - P(E)
Coin Toss	P(H) = P(T) = 1/2
Single Die	P(any number) = 1/6
Cards (total)	52 cards, 4 suits, 13 per suit

Solved Examples

Example 1: A coin is tossed 200 times and heads appear 108 times. Find P(heads) and P(tails).

Solution: P(heads) = 108/200 = 0.54. P(tails) = (200-108)/200 = 92/200 = 0.46.

Example 2: A die is rolled 300 times and 5 appears 48 times. Find P(getting 5).

Solution: P(5) = 48/300 = 0.16.

Example 3: From a deck of 52 cards, what is the probability of drawing a heart?

Solution: There are 13 hearts. P(heart) = 13/52 = 1/4.

Practice Questions

1. A coin is tossed 150 times and heads come up 78 times. Find P(heads) and P(tails).
2. A die is rolled 250 times. The number 6 appears 42 times. Find P(6).
3. What is the probability of getting a sum of 7 when two dice are rolled?
4. A bag contains 5 red and 3 blue balls. Find P(red) and P(blue).
5. From a deck of 52 cards, find the probability of drawing a king or a queen.

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