*The conditional probability of an event **A*, *given an event **B **with*:*Is defined by: *

This last equation specifies a new (conditional) probability law on the same sample space * Ω*. In particular, all properties of probability laws remain valid for conditional probability laws.

- Conditional probabilities can also be viewed as a probability law on a new universe
, because all of the conditional probability is concentrated on*B*.*B* - If the possible outcomes are finitely many and equally likely, then:

Explanation

Conditional probability provides us with a way to reason about the outcome of an experiment, based on partial information. Here are some examples of situations we have in mind:

- In an experiment involving two successive rolls of a die, you are told that the sum of the two rolls is 9. How likely is it that the first roll was a 6?
- In a word guessing game, the first letter of the word is a “t”. What is the likelihood that the second letter is “h”?
- How likely is it that a person has a certain disease given that a medical test was negative?
- A spot shows up on a radar screen. How likely is it to correspond to an aircraft?

In more precise terms, given an experiment, corresponding sample spaces, and a probability law, suppose that we know that the outcome is within some given event ** B**. We wish to quantify the likelihood that the outcome also belongs to some other given event

**. We thus seek to construct a new probability law that takes into account the available knowledge: a probability law that for any event**

*A***, specifies the conditional probability of**

*A***given**

*A***, denoted by**

*B**P(*.

**A**I**B**)An appropriate definition of conditional probability when all outcomes are equally likely is given by:

Generalizing the argument, we introduce the following definition of conditional probability:Where we assume that:

The conditional probability is undefined if conditioning event has zero probability. In words, out of the total probability of the elements of ** B**,

*P(*is the fraction that is assigned to possible outcomes that also belong to

**A**I**B**)**.**

*A*Sources:

- Introduction to probability (bertsekas, 2nd, 2008)
- Probability – The Science of Uncertainty and Data (MITx – 6.431x)

Literature review made by:

Prof. Larry Francis Obando – Technical Specialist – **Educational Content Writer**

WhatsApp: +34 633129287 +593998524011 **Atención Inmediata!!**

Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, * UCV *CCs

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, ** USB **Valle de Sartenejas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contacto: España +34 633129287

Caracas, Quito, Guayaquil, Cuenca.

WhatsApp: +34 633129287 +593998524011

FACEBOOK: DademuchConnection

email: dademuchconnection@gmail.com