Conditional probability

Conditional probabilities are useful when the occurrence of one event leads to the occurrence of another. If we have two events, A and B, where B has occurred and we want to find the probability of A occurring, we write this as follows:

Here, .

However, if the two events, A and B, are independent, then we have the following:

Additionally, if , then it is said that B attracts A. However, if A attracts B^C, then it repels B. 

The following are some of the axioms of conditional probability:

• _.
• _.
• _.
• is a probability function that works only for subsets of B. 
• _.
• If , then _.

The following equation is known as Bayes' rule:

This can also be written as follows:

Here, we have the following: 

• is called the prior.
• is the posterior.

• is the likelihood.
• acts as a normalizing constant.

The   symbol is read as  proportional to .

Often, we end up having to deal with complex events, and to effectively navigate them, we need to decompose them into simpler events. 

This leads us to the concept of partitions. A partition is defined as a collection of events that together makes up the sample space, such that, for all cases of B_i, .

In the coin flipping example, the sample space is partitioned into two possible events—heads and tails. 

If A is an event and B_i is a partition of Ω, then we have the following:

We can also rewrite Bayes' formula with partitions so that we have the following:

Here, .