Statstics

Exploring the world of Probability and its Applications

Published on August 23, 2023
360 Admin

Table of Contents

1 1. Classical Probability:
2 2. Statistical (Frequentist) Probability:
3 3. Axiomatic Probability:
4 Conditional Probability:
5 Bayes’ Theorem:
6 Applications in Real Life:

1. Classical Probability:

Classical probability is based on the assumption of equally likely outcomes in a sample space. It’s often used for situations where all possible outcomes are equally likely. The probability of an event A is calculated as:

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

Example: Rolling a fair six-sided die. The probability of rolling a 3 is $P(\text{rolling a 3}) = \frac{1}{6}$ .

2. Statistical (Frequentist) Probability:

The statistical probability is based on observations and data collected from real-world events. The probability of an event A is calculated as the relative frequency of A occurring in a large number of trials:

$P(A) = \lim_{n \to \infty} \frac{\text{Number of times A occurred}}{n}$

Example: Tossing a coin many times and observing the ratio of heads to total tosses.

3. Axiomatic Probability:

Axiomatic probability, also known as probability theory, is a mathematical framework that defines probability using a set of axioms. It formalizes probability as a measure of a probability space. The probability of an event A is denoted by $P(A)$ and satisfies the axioms of probability theory.

Example: The probability of drawing a red card from a standard deck is $P(\text{red card}) = \frac{26}{52}$ .

Conditional Probability:

Conditional probability is the probability of an event occurring given that another event has already occurred. It’s denoted as $P(A|B)$ and is calculated as:

$P(A|B) = \frac{P(A \cap B)}{P(B)}$

Example: Given that it’s raining ( $B$ ), what’s the probability of having an umbrella ( $A$ )?

Laws of Addition and Multiplication:
Law of Addition (Union): For any two events A and B:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Law of Multiplication (Intersection): For any two independent events A and B:

$P(A \cap B) = P(A) \cdot P(B)$

Independent Events:
Events A and B are independent if the occurrence of one event doesn’t affect the probability of the other event. Mathematically:

$P(A \cap B) = P(A) \cdot P(B)$

Example: Rolling a die and flipping a coin are typically independent events.

Theorem of Total Probability:
For a partition $B_1, B_2, \ldots, B_n$ of the sample space S:

$P(A) = \sum_{i=1}^{n} P(A \cap B_i) = \sum_{i=1}^{n} P(A|B_i) \cdot P(B_i)$

Bayes’ Theorem:

Bayes’ theorem relates conditional probabilities. For events A and B with $P(B) > 0$ :

$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$

Example: Spam Email Detection

Let:
– $A$ : Email is spam.
– $B$ : Email contains the word “discount.”

Probabilities:
– $P(A)$ : Probability that an email is spam (prior probability).
– $P(B|A)$ : Probability that an email contains the word “discount” given that it’s spam (likelihood).
– $P(B|\neg A)$ : Probability that an email contains the word “discount” given that it’s not spam (false positive rate).
– $P(\neg A)$ : Probability that an email is not spam (complementary probability to spam).

Bayes’ Theorem:

$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$

Using the law of total probability:

$P(B) = P(B|A) \cdot P(A) + P(B|\neg A) \cdot P(\neg A)$

Substituting into Bayes’ Theorem:

$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B|A) \cdot P(A) + P(B|\neg A) \cdot P(\neg A)}$

Applications in Real Life:

Probability theory finds applications in various fields such as finance, genetics, insurance, and more. For instance:

Weather Forecasting: Predicting the likelihood of rain based on historical weather data and atmospheric conditions.
Financial Modeling: Estimating the risk associated with different investment options.
Epidemiology: Analyzing the spread of diseases in a population and assessing the probability of an outbreak.
Machine Learning: Training models for tasks like image recognition, where probabilities help determine the most likely classification.