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@ -31,10 +31,15 @@ To store two coin flips, you might have the first subdivision represent the outc
| $0.25 - 0.5$ | Tails, Heads | | $0.25 - 0.5$ | Tails, Heads |
| $0.50 - 0.75$ | Heads, Tails | | $0.50 - 0.75$ | Heads, Tails |
| $0.75 - 1.00$ | Heads, Heads | | $0.75 - 1.00$ | Heads, Heads |
Imagine a situation where we want to store the coin flip *Heads, Heads, Tails*. Imagine a situation where we want to store all possible outcomes of three consecutive coin flips *Heads, Heads, Tails*.
Encoding this would happen as follows: Encoding this would happen as follows:
1. First we subdivide the range by the probability of each event happening. The probability of each is 50%, so that's simple. Referring above, we know that heads is represented by the top half of the range, and tails is represented by the bottom half of the range. 1. First we subdivide the range by the probability of each event happening. The probability of each is 50%, so that's simple. Referring above, we know that heads is represented by the top half of the range, and tails is represented by the bottom half of the range.
> Because the *first* coin flip resulted in *Heads*, the output value must be between $0.50$ and $1.00$. > Because the *first* coin flip resulted in *Heads*, the output value must be between $0.50$ and $1.00$.
2. Subdividing the range $0.50$ and $1.00$ again to store the results of the second flip, we end up with values between $0.50$ and $0.75$ representing the sequence *Heads, Tails*, and values between $0.75$ and $1.00$ representing the sequence *Heads, Heads*. 2. Subdividing the range $0.50$ and $1.00$ again to store the results of the second flip, we end up with values between $0.50$ and $0.75$ representing the sequence *Heads, Tails*, and values between $0.75$ and $1.00$ representing the sequence *Heads, Heads*.
> Because the *second* coin flip resulted in *Heads*, we know that the output value must be between $0.75$ and $1.00$ > Because the *second* coin flip resulted in *Heads*, we know that the output value must be between $0.75$ and $1.00$
3. Subdividing the range $0.75$ and $1.00$ yet again, $0. 3. Subdividing the range $0.75$ and $1.00$ yet again, $0.750$ - $0.875$ means the third coin flip resulted in *Tails*, and a value in the range $0.875$ - $1.000$ means the third coin flip resulted in *Heads*
> Because the *third coin flip resulted in *Heads*, any value between $0.875$ and $1.000$ encodes the fact that the first three coin flips went *Heads, Heads, Tails*.
The decoding process performs the same series of steps, but by asking a question instead of outputting a value.
1. Is the value between $0.00$ and $0.50$? If so, the first coin flip resulted in *Tails*. Otherwise if the value is between $0.50$ and $1.00$, the first coin flip resulted in *Heads*.
The above process can be repeated just like the encoding process until we've determined the output of the first three