Probability Formulas – Definition and Examples

Definition

Probability measures the likelihood of an event occurring and is a fundamental concept in statistics and real-world decision-making. It is represented as a number between 0 and 1, where 0 means an impossible event and 1 means a certain event.

Probability is the mathematical framework for quantifying uncertainty and measuring the likelihood of events occurring. It provides the foundation for statistical inference, decision making under uncertainty, and modeling random phenomena in science, engineering, and everyday life.

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Basic Probability Definition

Classical definition of probability for equally likely outcomes:

\[ P(A) \in [0,1] \quad \text{(Probability ranges between 0 and 1)} \]

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

\[ P(A) = \frac{|A|}{|S|} \]

\[ \text{Where: } A \text{ is the event, } S \text{ is the sample space} \]

\[ \text{Example: } P(\text{heads on coin}) = \frac{1}{2} = 0.5 \]

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Probability Axioms (Kolmogorov)

Fundamental axioms that define probability mathematically:

\[ \text{Axiom 1: } P(A) \geq 0 \quad \text{(Non-negativity)} \]

\[ \text{Axiom 2: } P(S) = 1 \quad \text{(Normalization)} \]

\[ \text{Axiom 3: } P(A_1 \cup A_2 \cup \ldots) = P(A_1) + P(A_2) + \ldots \]

\[ \text{for mutually exclusive events } A_i \text{ (Countable additivity)} \]

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Basic Probability Rules

Fundamental rules derived from the axioms:

\[ P(\emptyset) = 0 \quad \text{(Probability of impossible event)} \]

\[ P(A^c) = 1 - P(A) \quad \text{(Complement rule)} \]

\[ 0 \leq P(A) \leq 1 \quad \text{(Range of probability)} \]

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \quad \text{(Addition rule)} \]

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Union and Intersection of Events

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

\[ P(A \cap B) = P(A)P(B) \quad \text{(if A and B are independent)} \tag{1} \]

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Conditional Probability

Probability of one event given that another has occurred:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \quad \text{where } P(B) > 0 \]

\[ P(A \cap B) = P(A|B) \cdot P(B) = P(B|A) \cdot P(A) \]

\[ \text{Multiplication rule for dependent events} \]

\[ \text{Example: } P(\text{rain tomorrow}|\text{cloudy today}) \]

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Conditional Probability

Probability of one event given that another has occurred:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \quad \text{where } P(B) > 0 \]

\[ P(A \cap B) = P(A|B) \cdot P(B) = P(B|A) \cdot P(A) \]

\[ \text{Multiplication rule for dependent events} \]

\[ \text{Example: } P(\text{rain tomorrow}|\text{cloudy today}) \]

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Independence

When events do not influence each other:

\[ \text{Events A and B are independent if: } P(A \cap B) = P(A) \cdot P(B) \]

\[ \text{Equivalently: } P(A|B) = P(A) \text{ and } P(B|A) = P(B) \]

\[ \text{For multiple events: } P(A_1 \cap A_2 \cap \ldots \cap A_n) = P(A_1) \cdot P(A_2) \cdot \ldots \cdot P(A_n) \]

\[ \text{Example: Coin flips are independent events} \]

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Bayes' Theorem

Fundamental theorem for updating probabilities with new information:

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

\[ P(A_i|B) = \frac{P(B|A_i) \cdot P(A_i)}{\sum_{j} P(B|A_j) \cdot P(A_j)} \]

\[ \text{Where: } P(A) \text{ is prior, } P(A|B) \text{ is posterior} \]

\[ P(B|A) \text{ is likelihood, } P(B) \text{ is evidence} \]

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Law of Total Probability

Computing probability by conditioning on a partition:

\[ P(B) = \sum_{i} P(B|A_i) \cdot P(A_i) \]

\[ \text{Where } A_1, A_2, \ldots, A_n \text{ partition the sample space} \]

\[ P(B) = P(B|A) \cdot P(A) + P(B|A^c) \cdot P(A^c) \]

\[ \text{Two-event case with complements} \]

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Random Variables

Functions that assign numerical values to outcomes:

\[ X: S \rightarrow \mathbb{R} \quad \text{(Random variable as function)} \]

\[ P(X = x) = P(\{\omega \in S : X(\omega) = x\}) \]

\[ P(a \leq X \leq b) = \sum_{x: a \leq x \leq b} P(X = x) \quad \text{(Discrete)} \]

\[ P(a \leq X \leq b) = \int_a^b f_X(x) dx \quad \text{(Continuous)} \]

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Probability Distributions

Mathematical descriptions of random variable behavior:

\[ \text{PMF (Discrete): } p_X(x) = P(X = x) \]

\[ \text{PDF (Continuous): } f_X(x) \text{ where } P(X = x) = 0 \]

\[ \text{CDF: } F_X(x) = P(X \leq x) \]

\[ \sum_x p_X(x) = 1 \quad \text{or} \quad \int_{-\infty}^{\infty} f_X(x) dx = 1 \]

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Expected Value and Variance

Central measures of probability distributions:

\[ E[X] = \sum_x x \cdot P(X = x) \quad \text{(Discrete)} \]

\[ E[X] = \int_{-\infty}^{\infty} x \cdot f_X(x) dx \quad \text{(Continuous)} \]

\[ \text{Var}(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2 \]

\[ \text{Standard Deviation: } \sigma_X = \sqrt{\text{Var}(X)} \]

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Common Probability Distributions

Important probability models for different scenarios:

\[ \text{Bernoulli: } P(X = 1) = p, P(X = 0) = 1-p \]

\[ \text{Binomial: } P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

\[ \text{Poisson: } P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \]

\[ \text{Normal: } f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]

🎯 What does this mean?

Probability is the mathematical language for uncertainty - it gives us precise ways to measure and calculate how likely things are to happen. Think of it as the "mathematics of maybe" that turns vague statements like "probably" or "unlikely" into exact numbers between 0 and 1. It's the foundation for making rational decisions when we can't be certain about outcomes, from weather forecasting to medical diagnosis to financial planning.

\[ P(A) \]

Probability of Event A - Likelihood between 0 and 1

\[ S \]

Sample Space - Set of all possible outcomes

\[ A^c \]

Complement of A - Event A does not occur

\[ A \cap B \]

Intersection - Both events A and B occur

\[ A \cup B \]

Union - Either event A or B (or both) occurs

\[ P(A|B) \]

Conditional Probability - Probability of A given B

\[ X \]

Random Variable - Function from outcomes to numbers

\[ E[X] \]

Expected Value - Average value of random variable

\[ \text{Var}(X) \]

Variance - Measure of spread around the mean

\[ f_X(x) \]

Probability Density Function - Continuous probability model

\[ F_X(x) \]

Cumulative Distribution Function - P(X ≤ x)

\[ \emptyset \]

Empty Set - Impossible event with probability 0

🎯 Essential Insight: Probability is the mathematical bridge between uncertainty and precision - it transforms qualitative uncertainty into quantitative measures that enable rational decision-making! 🎯

🚀 Real-World Applications

🏥 Medical Diagnosis & Treatment

Clinical Decision Making & Risk Assessment

Disease probability calculations, treatment effectiveness, diagnostic test accuracy, and patient risk stratification using Bayesian methods

💰 Finance & Investment

Risk Management & Portfolio Optimization

Market risk assessment, option pricing, portfolio diversification, credit scoring, and insurance premium calculations based on probability models

🎯 Machine Learning & AI

Predictive Modeling & Classification

Probabilistic algorithms, Bayesian networks, uncertainty quantification, and decision trees rely on probability theory foundations

🌡️ Weather & Climate Science

Forecasting & Environmental Modeling

Weather prediction models, climate change projections, natural disaster probability, and environmental risk assessment using stochastic processes

The Magic: Medicine: Uncertainty quantification → Better diagnosis, Finance: Risk modeling → Informed investments, AI: Probabilistic reasoning → Smart predictions, Weather: Stochastic models → Accurate forecasts

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Master the "Uncertainty Quantification" Method!

Before solving probability problems, identify the type of uncertainty and appropriate probability model:

Key Insight: Probability is the systematic way to measure uncertainty and make it manageable. Every probability problem involves a sample space, events, and assignments of likelihood values that follow mathematical rules!

💡 Why this matters:

🔋 Real-World Power:

Decision Making: Quantify risks and benefits for optimal choices
Prediction: Forecast future events with measurable confidence
Quality Control: Monitor processes and detect anomalies statistically
Scientific Inference: Draw conclusions from data with known uncertainty

🧠 Mathematical Insight:

Axioms provide rigorous foundation for all probability calculations
Conditional probability enables updating beliefs with new information
Random variables connect probability to real-world measurements

🚀 Practice Strategy:

1 Define Sample Space and Events 🎯

Identify all possible outcomes (sample space S)
Define events as subsets of sample space
Key insight: Clear definitions prevent calculation errors

2 Apply Fundamental Rules 📏

Use complement rule: P(A^c) = 1 - P(A)
Apply addition rule: P(A∪B) = P(A) + P(B) - P(A∩B)
Check independence: P(A∩B) = P(A)P(B)?

3 Handle Conditional Situations 🔗

Use conditional probability: P(A|B) = P(A∩B)/P(B)
Apply Bayes' theorem for updating probabilities
Use law of total probability for complex scenarios

4 Choose Appropriate Models 📊

Discrete vs continuous random variables
Select distribution based on data characteristics
Calculate expected values and variances for interpretation

When you see probability as the systematic method for measuring and managing uncertainty, mathematics becomes a powerful tool for making rational decisions in an uncertain world!

Memory Trick: "Probability = Precise Reasoning Over Bewildering Assumptions By Intelligent Logic Integration Through Yonder" - MEASURE: Quantify uncertainty, MODEL: Mathematical framework, MANAGE: Make better decisions

🔑 Key Properties of Probability

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Axiomatic Foundation

Built on Kolmogorov's three axioms

Non-negativity, normalization, countable additivity

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Conditional Structure

Enables updating probabilities with new information

Foundation for Bayesian reasoning and learning

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Distribution Framework

Random variables provide numerical representation

Enables statistical analysis and modeling

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Universal Applicability

Models uncertainty in any domain

Foundation for statistics, machine learning, physics

Universal Insight: Probability is the mathematical embodiment of "measured uncertainty" - it transforms vague notions of likelihood into precise, manipulable numerical quantities! 🎯

Range Rule: All probabilities are between 0 and 1 inclusive

Complement Rule: P(A) + P(A^c) = 1 always

Independence Test: P(A∩B) = P(A)×P(B) if and only if independent

Bayes' Update: Prior × Likelihood = Posterior (up to normalization)

Probability – Basic Concepts and Rules