Probability is a branch of mathematics that deals with the analysis of random phenomena. It provides a framework for predicting the likelihood of various outcomes in uncertain situations. Probability theory is fundamental to many fields, including statistics, finance, science, engineering, and artificial intelligence.

Key Concepts in Probability

1. Basic Definitions
   • Experiment: A process that leads to observable outcomes. For example, rolling a die or flipping a coin.
   • Outcome: A possible result of an experiment. For example, getting a “4” when rolling a die.
   • Sample Space (S): The set of all possible outcomes of an experiment. For example, the sample space for rolling a die is \(S={1,2,3,4,5,6}\)
   • Event: A subset of the sample space. For example, the event “rolling an even number” is \(E={2,4,6}\).
2. Probability of an Event
   • The probability of an event E is a measure of the likelihood that E will occur. It is denoted by P(E) and calculated as:\(P(E)=\frac{Number \: of \: favorable \: outcomes}{Total \: number \: of \: possible \: outcomes}\)
   • Probability values range from 0 to 1, where:
     • P(E)=0: The event is impossible.
     • P(E)=1: The event is certain.
     • 0<P(E)<1: The event is possible but not certain.
3. Types of Probability
   • Theoretical Probability: Based on reasoning or theory. For example, the probability of rolling a “3” on a fair die is \(\frac{1}{6}\)​.
   • Experimental Probability: Based on actual experiments or historical data. For example, if a coin is flipped 100 times and lands on heads 55 times, the experimental probability of heads is \(\frac{55}{100}\)​=0.55.
   • Subjective Probability: Based on personal judgment or experience. For example, a weather forecaster might say there is a 70% chance of rain tomorrow.
4. Rules of Probability
   • Complementary Rule: The probability that an event does not occur is:\(P(E′)=1−P(E)\)where\( E′\) is the complement of E.
   • Addition Rule: For two events AA and BB, the probability of either A or B occurring is:\(P(A∪B)=P(A)+P(B)−P(A∩B)\)
     If A and B are mutually exclusive (cannot occur together), then:\(P(A∪B)=P(A)+P(B)\)
   • Multiplication Rule: For two independent events A and B, the probability of both A and B occurring is:
     \(P(A∩B)=P(A)⋅P(B)\)
     If A and B are dependent, then:
     \(P(A∩B)=P(A)⋅P(B∣A)\) where \(P(\frac{B}{A})\) is the probability of B given that A has occurred.
5. Conditional Probability
   • Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted by \(P(B∣A)\) and calculated as:\(P(B∣A)=\frac{P(A∩B)}{P(A)}\)​
   • Example: In a deck of cards, the probability of drawing a heart given that the card is red is:
     \(P(\frac{Heart}{Red})=P\frac{(Heart∩Red)}{P(Red)}\)
     =\(\frac{13/52}{26/52}=\frac{1}{2}\)
6. Independent and Dependent Events
   • Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other. For example, flipping a coin and rolling a die are independent events.
   • Dependent Events: Two events are dependent if the occurrence of one affects the probability of the other. For example, drawing two cards from a deck without replacement.
7. Bayes’ Theorem
   • Bayes’ Theorem is used to update the probability of an event based on new information. It is given by:
     \(P(A∣B)=\frac{P(B∣A)⋅P(A)}{P(B)}\)
   • Example: In medical testing, Bayes’ Theorem can be used to find the probability of having a disease given a positive test result.
8. Random Variables
   • A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. There are two types:
     • Discrete Random Variable: Takes on a countable number of values (e.g., number of heads in 3 coin flips).
     • Continuous Random Variable: Takes on an infinite number of values within a range (e.g., height or weight).
9. Probability Distributions
   • A probability distribution describes how probabilities are distributed over the values of a random variable.
   • Discrete Probability Distribution: Examples include the Binomial and Poisson distributions.
   • Continuous Probability Distribution: Examples include the Normal and Uniform distributions.
10. Expected Value and Variance
    • Expected Value (Mean): The long-run average value of a random variable. For a discrete random variable XX, it is calculated as:
      \(E(X)=∑x⋅P(x)\)
    • Variance: Measures the spread of the distribution. It is calculated as:
      \(Var(X)=E(X^2)−[E(X)]^2\)
11. Law of Large Numbers
    • As the number of trials in an experiment increases, the average of the results will converge to the expected value.
12. Central Limit Theorem
    • The Central Limit Theorem states that the sampling distribution of the mean of any independent, random variable will be approximately normal if the sample size is large enough.

Applications of Probability

• Statistics: Probability forms the foundation of statistical inference.
• Finance: Used in risk assessment and option pricing.
• Science and Engineering: Used to model uncertainty and randomness in systems.
• Artificial Intelligence: Used in machine learning algorithms and decision-making processes.