CBSE Class 10 Maths Notes: Probability

Definitions: Probability Unveiled

Probability is a measure of the likelihood that an event will occur. It’s a fundamental concept in mathematics and statistics, helping us quantify uncertainty. Essentially, it tells us “how likely” something is to happen. Probability values range from 0 to 1, where 0 indicates impossibility and 1 indicates certainty.

Key Terms:

• Experiment: An action or process that leads to well-defined outcomes.
• Outcome: A possible result of an experiment.
• Event: A specific set of outcomes of an experiment.
• Sample Space: The set of all possible outcomes of an experiment.

Core Principles: Classical Probability

The classical (or theoretical) definition of probability assumes all outcomes in the sample space are equally likely.

Formula:

The probability of an event, $P(E)$, is calculated as:

$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Where:

• Favorable outcomes are those that constitute the event E.
• The total number of possible outcomes is the size of the sample space.

Formulae & Calculations

Let’s break down some common probability scenarios with formulas:

Probability of an event not happening (complement):

$P(\text{not } E) = 1 – P(E)$

Probability of certain events:

$P(\text{certain event}) = 1$

Probability of impossible events:

$P(\text{impossible event}) = 0$

Examples: Rolling Dice, Flipping Coins & Drawing Cards

1. Flipping a Coin:

What is the probability of getting heads when flipping a fair coin?

Solution:

• Sample space: {Heads, Tails} (2 outcomes)
• Favorable outcome: Heads (1 outcome)
• $P(\text{Heads}) = \frac{1}{2} = 0.5$ or 50%

2. Rolling a Die:

What is the probability of rolling a 4 on a fair six-sided die?

Solution:

• Sample space: {1, 2, 3, 4, 5, 6} (6 outcomes)
• Favorable outcome: 4 (1 outcome)
• $P(\text{Rolling a 4}) = \frac{1}{6}$

3. Drawing a Card (Basic Case):

What is the probability of drawing an Ace from a standard deck of 52 playing cards?

Solution:

• Sample space: 52 cards (52 outcomes)
• Favorable outcome: 4 Aces (4 outcomes)
• $P(\text{Drawing an Ace}) = \frac{4}{52} = \frac{1}{13}$

Real-Life Examples: Probability in Action

Let’s see how probability pops up in everyday scenarios:

Example 1: Weather Forecasting

If a weather forecast predicts a 70% chance of rain, it’s expressing the probability of rain occurring.

Example 2: Insurance

Insurance companies use probability to estimate the likelihood of events (e.g., accidents, illnesses) to set premiums.

Example 3: Games of Chance

Games like lottery and other casino games are built upon probability. Players use it to understand their odds of winning.