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C3: Conditional Probability And Independence

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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C3: Conditional Probability And Independence

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  1. 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

  2. 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. 3.1 – Conditional Probability Provided P(C) > 0 3.2 – Multiplication Rule For any events A and C:

  4. 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:

  5. 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

  6. 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.

  7. 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.

  8. 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.

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