# Probability Theory

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Probability Theories in a very simple style with understandable examples.

### Introduction

**What is Probability?**

What we actually mean when we say that *the probability of a certain event A *is equal to *x.*

Mathematically, we can think about probability as a number that ranges between 0 and 1 and possesses certain properties. And empirically, we can think about a probability matching up with our intuition of the likelihood that a certain event occurs. When the probability of occurrence of an event is 0, it never happens, and an event having probability 1 is bound to happen.

**PROBABILITY FUNCTION**

Think about a *probability function as a function that *assigns numbers between 0 and 1 to events according to three defining properties.

RANDOM EXPERIMENT, SAMPLE SPACE, POSSIBLE OUTCOME

Probability starts with experiments whose outcome is uncertain. Uncertainty in the occurrence of outcomes is the defining property of probability. In the world where outcomes are certain, probability has no role to play.

A discrete sample space can be finite or infinite.

**Finite Discrete Sample Space:**

Ω = {ω1, . . . , ωk}

More concretely we can use a set of numbers that start with a number and ends.

Ω = {1, 2, 3….50}, it has a total of 50 numbers

**PROPERTIES OF PROBABILITIES**

We very often combine events together to create new events using the connectives “or,” (union) “and,” (intersection) and “not” (complement).

EQUALLY LIKELY OUTCOMES

The simplest probability model for a finite sample space is that all outcomes are equally likely. If Ω has k elements, then the probability of each outcome is 1/k, since probabilities sum to 1. That is, P(ω) = 1/k, for all ω ∈ Ω. Computing probabilities for equally likely outcomes takes a fairly simple form. Suppose A is an event with s elements, with s ≤ k. Since P(A) is the sum of the probabilities of all the outcomes contained in A,

*MONTE CARLO SIMULATION*

Using random numbers on a computer to simulate probabilities is called the Monte Carlo method. The name was coined in the 1940s by mathematicians **John von Neumann** and **Stanislaw Ulam** working on the Manhattan Project. It was named after the famous Monte Carlo casino in Monaco.