Section 2

All Probability is Conditional Probability (BH Chapter 2)

Conditional Probability - Suppose we observe event B and are interested in the probability of event A occurring given this information. Then,

Bayes’ Rule - This is arguably one of the most important concepts and tools you will learn in this course.

Bridging Conditional Probability and Sets

An intuitive way to visualize conditional probability is to think about the intersection of sets. In order to find the intersection of two different sets A and B, we establish one of these sets to be our sample space and find the likely occurrence of the other set within this established sample space.

Law of Total Probability (LOTP)

Extra Conditioning

Disjoint vs. Independent

Independent events are events such that observing event B yields no information about the possibility of also observing event A. That is, conditioning on observing event B, the probability of observing event A is unchanged:

We can apply this result to Bayes’ rule and quickly demonstrate an alternative definition of independence:

However, just as pairwise independence does not imply conditional independence, conditional independence does not imply pairwise independence.

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