Probability Chapter Notes | Mathematics Class 10 ICSE PDF Download

Introduction

Imagine you're flipping a coin, rolling a die, or picking a card from a deck. Will it be heads or tails? A 6 or a 1? A king or an ace? Life is full of such uncertainties, and probability is the exciting branch of mathematics that helps us measure and understand these uncertainties. Whether you're guessing the chance of rain this afternoon or predicting the outcome of a game, probability gives us a way to make sense of the unknown. It’s like a magical tool that turns vague phrases like "maybe" or "probably" into precise numbers, making it both fun and incredibly useful in everyday decisions!

• Probability is the study of how likely something is to happen.
• We often use words like "most likely," "probably," or "no chance" in daily life, which show uncertainty.
• Probability measures this uncertainty with numbers.
• It is both mathematically interesting and practically important.

Some Basic Terms and Concepts

1. Experiment

• An experiment is any process that gives a clear, well-defined result.
• It is a planned action with specific possible outcomes.
• Steps to understand:
  • Perform an action (like tossing a coin).
  • Observe the result (head or tail).
  • The result is always clear and defined.

2. Random Experiment

A random experiment is an experiment where all possible outcomes are known, but the exact outcome is unpredictable.
Steps to identify:

• List all possible outcomes (known in advance).
• Realize that you cannot predict which one will happen.

It can have two or more outcomes.

3. Sample Space

The sample space is the set of all possible outcomes of a random experiment, denoted by S.
Steps to determine:

• Identify the experiment.
• List every possible outcome.
• Write them as a set.

4. Equally Likely Outcomes

Outcomes are equally likely if each has the same chance of occurring.
Steps to check:

• Ensure all outcomes are known in advance.
• Confirm each outcome has an equal chance of happening.

Not all experiments have equally likely outcomes.

5. An Event

• An event is any outcome or set of outcomes in a random experiment.
• It represents something that happens during the experiment.
• Steps to define:
  • Conduct the random experiment.
  • Identify specific outcomes or groups of outcomes.
  • Call these outcomes an event.

Measurement of Probability

• Probability measures how likely an event is to happen.
• It is calculated as a ratio of favorable outcomes to total outcomes.
• Formula: P(E) = Number of favorable outcomes / Total number of possible outcomes
• Steps to calculate:
  • Identify the total number of possible outcomes (n).
  • Count the number of favorable outcomes for the event (m).
  • Divide m by n to get P(E) = m/n.

1. Empirical (or Experimental) Probability

• Empirical probability is based on actual experiments or observations.
• It requires recording outcomes from repeated trials.
• It is only an estimate, as results may vary in different experiments.
• Steps to calculate:
  • Perform the experiment multiple times.
  • Record the number of times the event occurs.
  • Divide by the total number of trials.

2. Classical (or Theoretical) Probability

• Classical probability is calculated without performing the experiment, assuming outcomes are equally likely.
• It uses logical reasoning based on the sample space.
• Formula: P(E) = Number of favourable outcomes / Number of all possible outcomes
• Steps to calculate:
  • Determine the sample space.
  • Count favourable outcomes for the event.
  • Divide to find the probability.
• In this chapter, probability refers to classical probability.

Elementary Event

An elementary event is an event with only one favorable outcome.
Steps to identify:

• List all outcomes in the sample space.
• Check if the event has exactly one favourable outcome.

Complementary Events

• Complementary events are two events where one is the occurrence of an event (E) and the other is its non-occurrence (not E, denoted E̅).
• Formula: P(E) + P(E̅) = 1
• P(E̅) = 1 - P(E)
• Steps to find:
  • Calculate P(E) using favourable and total outcomes.
  • Subtract P(E) from 1 to get P(E̅).
• The sum of the probabilities of an event and its complementary event is always 1.

Impossible and Sure Events

• An impossible event has a probability of 0 (cannot happen).
• A sure event has a probability of 1 (will definitely happen).
• Range of probability: 0 ≤ P(E) ≤ 1
• Steps to identify:
  • If no outcomes favor the event, it’s impossible (P(E) = 0).
  • If all outcomes favor the event, it’s sure (P(E) = 1).

Important

Tossing a coin once:

• Total outcomes: 2 (Head, Tail).
• Number of outcomes = 2¹ = 2.

Tossing a coin two times:

• Total outcomes: 4 (HH, HT, TH, TT).
• Number of outcomes = 2² = 4.

Tossing a coin three times:

• Total outcomes: 8 (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT).
• Number of outcomes = 2³ = 8.
• Tossing multiple coins gives the same outcomes whether done simultaneously or one at a time.