Home > Articles > Networking

  • Print
  • + Share This
This chapter is from the book

1.2. Joint and Conditional Probability

Thus far, we have defined the terms used in studying probability and considered single events in isolation. Having set this foundation, we now turn our attention to the interesting issues that arise when studying sequences of events. In doing so, it is very important to keep track of the sample space in which the events are defined: A common mistake is to ignore the fact that two events in a sequence may be defined on different sample spaces.

1.2.1. Joint Probability

Consider two processes with sample spaces S1 and S2 that occur one after the other. The two processes can be viewed as a single joint process whose outcomes are the tuples chosen from the product space S1 × S2. We refer to the subsets of the product space as joint events. Just as before, we can associate probabilities with outcomes and events in the product space. To keep things straight, in this section, we denote the sample space associated with a probability as a subscript, so that PS1(E) denotes the probability of event E defined over sample space S1, and PS1 × S2 (E) is an event defined over the product space S1 × S2.

We will return to the topic of joint processes in Sections 1.8. We now turn our attention to the concept of conditional probability.

1.2.2. Conditional Probability

Common experience tells us that if a sky is sunny, there is no chance of rain in the immediate future but that if the sky is cloudy, it may or may not rain soon. Knowing that the sky is cloudy, therefore, increases the chance that it may rain soon, compared to the situation when it is sunny. How can we formalize this intuition?

To keep things simple, first consider the case when two events E and F share a common sample space and occur one after the other. Suppose that the probability of E is PS(E) and the probability of F is PS (F). Now, suppose that we are informed that event E actually occurred. By definition, the conditional probability of the event F conditioned on the occurrence of event E is denoted PS × S(F|E) (read “the probability of F given E”) and computed as

If knowing that E occurred does not affect the probability of F, E and F are said to be independent and

PS × S (EF) = PS(E)PS(F)

The notion of conditional probability generalizes to the case in which events are defined on more than one sample space. Consider a sequence of two processes with sample spaces S1 and S2 that occur one after the other. (This could be the condition of the sky now, for instance, and whether it rains after 2 hours.) Let event E be a subset of S1 and event F a subset of S2. Suppose that the probability of E is PS1(E) and the probability of F is PS2(F). Now, suppose that we are informed that event E occurred. We define the probability PS1 × S2 (F|E) as the conditional probability of the event F conditional on the occurrence of E as

01equ05.jpg

If knowing that E occurred does not affect the probability of F, E and F are said to be independent and

01equ06.jpg

It is important not to confuse P(F|E) and P(F). The conditional probability is defined in the product space S1 × S2 and the unconditional probability in the space S2. Explicitly keeping track of the underlying sample space can help avoid apparent contradictions such as the one discussed in Example 1.14.

1.2.3. Bayes’s Rule

One of the most widely used rules in the theory of probability is due to an English country minister: Thomas Bayes. Its significance is that it allows us to infer “backwards” from effects to causes rather than from causes to effects. The derivation of his rule is straightforward, though its implications are profound.

We begin with the definition of conditional probability (Equation 1.4):

011equ01.jpg

If the underlying sample spaces can be assumed to be implicitly known, we can rewrite this as

01equ07.jpg

We interpret this to mean that the probability that both E and F occur is the product of the probabilities of two events: first, that E occurs; second, that conditional on E, F occurs.

Recall that P(F|E) is defined in terms of the event F following event E. Now, consider the converse: F is known to have occurred. What is the probability that E occurred? This is similar to the problem: If there is fire, there is smoke, but if we see smoke, what is the probability that it was due to a fire? The probability we want is P(E|F). Using the definition of conditional probability, it is given by

01equ08.jpg

Substituting for P(F) from Equation 1.7, we get

01equ09.jpg

which is Bayes’s rule. One way of interpreting this is that it allows us to compute the degree to which some effect, or posterior F, can be attributed to some cause, or prior E.

We can generalize Bayes’s rule when a posterior can be attributed to more than one prior. Consider a posterior F that is due to some set of n priors Ei such that the priors are mutually exclusive and exhaustive: That is, at least one of them occurs, and only one of them can occur. This implies that 012fig01.jpg. Then,

This is also called the law of total probability.

The law of total probability allows one further generalization of Bayes’s rule to obtain Bayes’s theorem. From the definition of conditional probability, we have

012equ02.jpg

From Equation 1.7, we have

013equ01.jpg

Substituting Equation 1.10, we get

01equ11.jpg

This is called the generalized Bayes’s rule, or Bayes’s theorem. It allows us to compute the probability of any one of the priors Ei, conditional on the occurrence of the posterior F. This is often interpreted as follows: We have some set of mutually exclusive and exhaustive hypotheses Ei. We conduct an experiment, whose outcome is F. We can then use Bayes’s formula to compute the revised estimate for each hypothesis.

  • + Share This
  • 🔖 Save To Your Account

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

Overview


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information


To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

Newsletters

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information


Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.

Security


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

Children


This site is not directed to children under the age of 13.

Marketing


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information


If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

Choice/Opt-out


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information


Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents


California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure


Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

Links


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact


Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice


We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020