PROBABILITY

Probability is a concept used to quantify the likelihood of an event occurring. It's expressed as a number between 0 and 1, with 0 meaning the event is impossible and 1 meaning it's certain. There are several ways to understand and categorize probability, each with its own advantages and applications. Here are some of the most common types:

1. Classical Probability:

• Based on equally likely outcomes in a finite sample space.
• Assumes all possible outcomes have an equal chance of occurring, making calculations straightforward.
• Used in games of chance like rolling dice or flipping coins.

2. Frequentist Probability:

• Based on the relative frequency of an event in repeated trials.
• Estimates the probability of an event by observing how often it happens over a large number of repetitions.
• Used in scientific experiments, polls, and statistical analysis.

3. Bayesian Probability:

• Based on updating beliefs based on new evidence using Bayes' theorem.
• Incorporates prior knowledge about an event and revises it as new information becomes available.
• Used in machine learning, medical diagnosis, and risk assessment.

4. Subjective Probability:

• Based on personal beliefs and judgments about the likelihood of an event.
• Reflects an individual's degree of confidence in an event occurring, often based on experience or intuition.
• Used in decision-making under uncertainty and assessing subjective risk.

5. Axiomatic Probability:

• Based on a set of axioms that define the properties of probability.
• Provides a mathematical framework for working with probability without relying on specific interpretations.
• Used in theoretical probability and advanced mathematical models

Formulas of Probability

Type of Probability	Formula	Description
Classical Probability	P(E) = n(E) / n(S)	Calculates the probability (P) of event (E) occurring, where n(E) is the number of favorable outcomes and n(S) is the total number of outcomes in the sample space (S).
Frequentist Probability	P(E) = f(E)	Estimates the probability (P) of event (E) by observing its frequency (f) in repeated trials.
Bayesian Probability:	P(H	E) = P(E
Subjective Probability	P(E) = personal degree of belief	Represents an individual's personal assessment of the likelihood of event (E) occurring, often based on experience or intuition.
Conditional Probability:	P(A	B) = P(A ∩ B) / P(B)
Joint Probability:	P(A ∩ B) = P(A) × P(B) (if independent)	Calculates the probability (P) of both events (A) and (B) happening together, assuming they are independent (P(A) doesn't affect P(B) and vice versa).
Marginal Probability:	P(A) = Σ P(A ∩ Bi) for all Bi in S	Calculates the probability (P) of event (A) occurring regardless of other events (Bi) in the sample space (S).
Independent Events:	P(A ∩ B) = P(A) × P(B)	Applies when the occurrence of event (A) doesn't influence the probability of event (B) happening and vice versa.
Dependent Events:	P(A ∩ B) ≠ P(A) × P(B)	Represents situations where the occurrence of one event (A) affects the probability of the other event (B) happening.

 

 

 

​

​

Solved Problems of Probability   Problem 1: Classical Probability Question: In a standard deck of playing cards, what is the probability of drawing a red card? Solution: A standard deck has 52 cards, and half of them are red. Therefore, the probability ($P$) of drawing a red card is given by: (Red Card)=Total Number of Cards/Number of Red Cards​=26/52​=1/2 Problem 2: Conditional Probability Question: A bag contains 5 red balls and 3 green balls. If two balls are drawn without replacement, what is the probability that the second ball is green given that the first ball is red? Solution: Initially, there are 8 balls in the bag. After drawing a red ball, there are 7 balls left. The probability ($P$) of drawing a green ball given that the first ball is red is given by: P(Green∣Red)=Total Balls left/Number of Green Balls left​=3/7