Conditional Probability

DEFINITION: Given a a Sample Space $\QTR{Large}{S}$ , a Sigma Algebra $\QTR{Large}{A}$ , a Probability Measure $\QTR{Large}{P}$ on $\QTR{Large}{A}$ , and Events and in $\QTR{Large}{A.}$ Assuming

define the Conditional Probability of given

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An Example:

The Events:

$\QTR{Large}{S~=~}$ {The days of October and November 2012}

$\QTR{Large}{O~=~}$ {The days of October 2012}

$\QTR{Large}{N~=~}$ {The days of November 2012}

The various days of the week in $\QTR{Large}{S}$

$\QTR{Large}{R~=~}$ {The days it rained in $\QTR{Large}{S}$ }

Finally, $\QTR{Large}{A}$ is the Sigma Algebra generated by the given Events.

The Probabilities:

Most are arithmetic;

, , ,

, $\bigskip$

From weather data:

From Calculations:

Since and $\$ are disjoint,

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Definition:

Events and are said to be Independent if or equivalently or equivalently

Examples:

The Experiment is flip a coin twice is get a Head the first time and is get a Head the second time .
The Experiment is choose a card from a standard deck, is the card is a three and the card is a heart.

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Bayes Rule:

since

What is the probability that it was a day in October given that we know it rained on that day?

An Aside: The Rule present above is special case of a more general Bayes Formula of that will come into play later in the semester. I will discuss it at that time but at this point it would be worthwhile to at least review the material in Grinstead and Snell's Introduction to Probability beginning on Page 145.

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A more relevent example:

We are given a communications channel that transmits $\QTR{Large}{1}$ s and $\QTR{Large}{0}$ s with conditional probabilities , , ,.

That is, for example, given that we transmit a $\QTR{Large}{1}$ the probability that a $\QTR{Large}{1}$ will be received is .

Question: Assume that we know that given that we receive a $\QTR{Large}{1}$ what is the probability that a $\QTR{Large}{1}$ was transmitted., compute ?

and

MATH

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Exercise, The not so wonderful strategy:

Assume . Fill in the question marks, using the previous diagram.

Finally compute