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notes/ANS Theory.md

@@ -17,4 +17,20 @@ Arithmetic coding works by taking a stream of data, and converting it into an in
 
 For example, the probability of a coin flip resulting in tails is 50%, and the probability of a coin flip resulting in heads is 50%. The probability of a coin flip resulting in heads *or* tails is %100.
 
-If we wanted to keep track of the result of a series of coin flips, this could be done by subdividing a range. If the coin flip is between $0$ and $0.5$, then we know that the first flip must 
+If we wanted to keep track of the result of a series of coin flips, this could be done by subdividing a range. If the coin flip is between $0$ and $0.5$, then we know that the first flip must have been tails.
+
+If the coin flip is between $0.5$ and $1$, then we know that the first flip must have been heads.
+
+This subdivision process can be repeated infinitely to store an infinite number of coin flips by dividing each range again.
+
+To store two coin flips, you might have the first subdivision represent the outcome of the first coin flip, and the second subdivision represent the outcome of the second coin flip:
+
+| Range         | Result       |
+| ------------- | ------------ |
+| $0.00 - 0.25$ | Tails, Tails |
+| $0.25 - 0.5$  | Tails, Heads |
+| $0.50 - 0.75$ | Heads, Tails |
+| $0.75 - 1.00$ | Heads, Heads |
+Imagine a situation where we want to store the coin flip *Heads, Heads, Tails*. 
+1. First we subdivide the range by the probability of each event happening. The probability of each is 50%, so that's simple. 
+2. We know that the r