1 minute read

Conditional Probability (Atlast)

Finally got the basic intuition for conditional probabilities to click. All thanks to Statlect . The way to look at conditional probabilities it so simply consider them as subsetting our sample space. Consider 2 events: E and I. If I simply want to consider the probability of E then for a uniform probability law I would simply consider the cardinalities i.e

\[P(E) = \frac{card(E)}{card(\Omega)} \\ ..\\ P(I) = \frac{card(I)}{card(\Omega)}\]

If I was then given some additional information that told me that the event I has occured and I wanted to know what is the probability that event E will occur. This means that I am now interested in the event E that is a subset of I. This is written as

\[P(E|I)\]

In terms of cardinalities we can interpret the probability of the event E given I to be the elements of E that are in I divided by the total elements of I. We can then convert the cardinalities to probabilities by dividing the numerator and denominator by the cardinality of the sample space

\[P(E|I) = \frac{card(E\cap I)}{card(I)} \\. \\ P(E|I) = \frac{card(E\cap I)/card(\Omega)}{card(I)/card(\Omega)} \\ . \\ P(E|I) = \frac{P(E\cap I)}{P(I)}\] 
Apparently I can separate a paragraph into columns by 
inserting a pipe | like this.
Test sentence before a pipe. Test sentence before a pipe. Test sentence before a pipe.Test sentence before a pipe. Test sentence before a pipe Test sentence after the pipe. Test sentence after the pipe. Test sentence after the pipe. Test sentence after the pipe.