C3: Conditional Probability And Independence

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MATH 3033 based on Dekking et al. A Modern Introduction to Probability and Statistics. 2007 Slides by James Connolly Instructor Longin Jan Latecki . C3: Conditional Probability And Independence. 3.1 – Conditional Probability.

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MATH 3033 based onDekking et al. A Modern Introduction to Probability and Statistics. 2007Slides by James Connolly Instructor Longin Jan Latecki

C3: Conditional Probability And Independence

3.1 – Conditional Probability
• Conditional Probability: the probability that an event will occur, given that another event has occurred that changes the likelihood of the event
• Example:If event L is “person was born in a long month”, and event R is “person was born in a month with the letter ‘R’ in it”, then P(R) is affected by whether or not L has occurred. The probability that R will happen, given that L has already happened is written as:

P(R|L)

3.1 – Conditional Probability

Provided P(C) > 0

3.2 – Multiplication Rule

For any events A and C:

3.3 – Total Probability & Bayes Rule

The Law of Total Probability

Suppose C1, C2, … ,CM are disjoint events such that

C1 U C2 U … U CM = Ω. The probability of an arbitrary

event A can be expressed as:

3.3 – Total Probability & Bayes Rule

Bayes Rule:

Suppose the events C1, C2, … CM are disjoint

and C1 U C2 U … U CM = Ω. The conditional probability of Ci, given an arbitrary event A, can be expressed as:

or

3.4 – Independence

Definition:

An event A is called independent of B if:

That is to say that A is independent of B if the probability of A occurring is not

changed by whether or not B occurs.

3.4 – Independence

Tests for Independence

To show that A and B are independent we have to prove just one of the following:

A and/or B can both be replaced by their complement.

3.4 – Independence

Independence of Two or More Events

Events A1, A2, …, Am are called independent if:

This statement holds true if any event or events is/are replaced by their

complement throughout the equation.