Basic Probability Concepts

Definition of Probability:
- The measure of the likelihood that an event will occur.
- Expressed as a number between 0 (impossible event) and 1 (certain event).
Probability Formula:
- ( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} )
Types of Events:
- Simple Event: An event with a single outcome.
- Compound Event: An event with two or more outcomes.
Sample Space (S):
- The set of all possible outcomes of a probability experiment.
- Denoted as ( S ).
Event (E):
- Any subset of the sample space.
Complement of an Event:
- The complement of event A (denoted as ( A' )) consists of all outcomes in the sample space that are not in A.
- Formula: ( P(A') = 1 - P(A) )
Union of Events:
- The union of events A and B (denoted as ( A \cup B )) includes all outcomes in A, in B, or in both.
- Formula:
  - For mutually exclusive events: ( P(A \cup B) = P(A) + P(B) )
  - For non-mutually exclusive events: ( P(A \cup B) = P(A) + P(B) - P(A \cap B) )
Intersection of Events:
- The intersection of events A and B (denoted as ( A \cap B )) includes outcomes that are in both A and B.
- Formula: ( P(A \cap B) )
Independent Events:
- Two events A and B are independent if the occurrence of one does not affect the occurrence of the other.
- Formula: ( P(A \cap B) = P(A) \cdot P(B) )
Dependent Events:
- Two events A and B are dependent if the outcome of one event affects the outcome of the other.
- Formula: ( P(A \cap B) = P(A) \cdot P(B|A) )
Conditional Probability:
- The probability of event A occurring given that event B has occurred.
- Denoted as ( P(A|B) ) and calculated using:
  - ( P(A|B) = \frac{P(A \cap B)}{P(B)} )
Law of Total Probability:
- If events B1, B2, ..., Bn partition the sample space, then for any event A:
  - ( P(A) = P(A \cap B1) + P(A \cap B2) + ... + P(A \cap Bn) )
Common Probability Distributions:
- Discrete Distributions: Focus on countable outcomes (e.g., Binomial, Poisson).
- Continuous Distributions: Focus on measurable outcomes (e.g., Normal, Exponential).

These concepts form the foundation of probability and are crucial for further study in statistics, data analysis, and various applied fields.

Probability Basics

Probability measures the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain).
Probability Formula: ( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} )

Types of Events and Sample Space

A simple event has a single outcome.
A compound event has multiple outcomes.
The sample space (S) encompasses all possible outcomes of a probability experiment. It is denoted by ( S ).
An event (E) is any subset of the sample space.

Complement of an Event

The complement of event A (A') includes all outcomes in the sample space that are not in A.
Formula: ( P(A') = 1 - P(A) )

Combining Events

The union of events (A ∪ B) encompasses outcomes in either A, B, or both.
- For mutually exclusive events: ( P(A \cup B) = P(A) + P(B) )
- For non-mutually exclusive events: ( P(A \cup B) = P(A) + P(B) - P(A \cap B) )
The intersection of events (A ∩ B) includes outcomes that are present in both A and B.
- Formula: ( P(A \cap B) )

Event Dependence and Independence

Independent events (A and B) occur when one event's outcome does not affect the other's outcome.
- Formula: ( P(A \cap B) = P(A) \cdot P(B) )
Dependent events (A and B) occur when one event's outcome does affect the other's outcome.
- Formula: ( P(A \cap B) = P(A) \cdot P(B|A) )

Conditional Probability and the Law of Total Probability

Conditional probability ( P(A|B) ) calculates the probability of event A happening given that event B has already occurred.
- Formula: ( P(A|B) = \frac{P(A \cap B)}{P(B)} )
The Law of Total Probability states that if events B1, B2,..., Bn partition the sample space, then for any event A:
- ( P(A) = P(A \cap B1) + P(A \cap B2) +...+ P(A \cap Bn) )

Probability Distributions

Discrete distributions focus on countable outcomes, such as the Binomial and Poisson distributions.
Continuous distributions focus on measurable outcomes, such as the Normal and Exponential distributions.

Probability Basics

Probability describes the likelihood of an event happening, ranging from 0 (impossible) to 1 (certain).

Key Terms

Experiment: Any process that leads to an outcome.
Sample Space (S): The complete set of all possible outcomes of an experiment.
Event: A specific outcome or collection of outcomes from the sample space.

Types of Events

Certain Event: An event that is guaranteed to happen (Probability = 1).
Impossible Event: An event that cannot happen (Probability = 0).
Simple Event: An event with only one outcome.
Compound Event: An event made up of two or more simple events.

Probability Formula

The probability of a simple event (P(E)) is calculated by dividing the number of favorable outcomes by the total number of outcomes in the sample space:

[ P(E) = \frac{\text{Number of favorable outcomes for event } E}{\text{Total number of outcomes in sample space S}} ]

Complementary Events

The complement of event A (denoted as A') includes all outcomes in the sample space that are not in A.
The sum of probabilities of an event and its complement always equals 1:

[ P(A') = 1 - P(A) ]

Addition Rule of Probability

Mutually Exclusive Events: Events that cannot happen at the same time.
For mutually exclusive events A and B:

[ P(A \cup B) = P(A) + P(B) ]
For any events A and B:

[ P(A \cup B) = P(A) + P(B) - P(A \cap B) ]

Multiplication Rule of Probability

Independent Events: Events where the occurrence of one does not affect the probability of the other.
For independent events A and B:

[ P(A \cap B) = P(A) \times P(B) ]

Applications of Probability

Probability is crucial in various fields:
- Statistics
- Finance
- Science and Engineering
- Games and Sports

Key Concepts for Class 10

Conditional Probability: The probability of event A occurring given that event B has already happened, denoted as P(A|B).

[ P(A|B) = \frac{P(A \cap B)}{P(B)} ] (if P(B) > 0)
Bayes' Theorem: A formula used to find conditional probabilities.
Law of Large Numbers: As the number of trials of an experiment increases, the experimental probability will approach the true (theoretical) probability.