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we begin with a problem

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Alice and Bob leave in tree forts

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which are far apart[br]with no line of sight between them

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and they need to communicate

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so they decide to run a wire[br]between the two houses

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they pull a wire tight

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and attach it tin can to each end

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allowing them to send their voices[br]faintly along the wire

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however there is a problem

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noise

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whenever there is a high wind

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it becomes impossible to hear[br]the signal over the noise

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so they need a way to increase[br]the energy level of the signal

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to separate it from the noise

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this gives bob an idea

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they can simply pluck the wire

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which is much easier to detect[br]over the noise

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but this leads to a new problem

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how do they encode their[br]messages as plucks?

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well since they want to play[br]board games across a distance

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they tackle the most common messages first

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the outcome of two dice rolls

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in this case the messages they are sending

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can be thought of as a selection[br]from a finite number of symbols

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in this case the 11 possible numbers

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which we call a discrete source

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at first they decide to use[br]the simplest method

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they send the result as[br]the number of plucks

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so to send a 3 they send three plucks

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9 is nine plucks and 12 is twelve plucks

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however they soon realize that this[br]takes much longer than it needs to

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from practice they find that their[br]maximum plucks speed is

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two plucks per second

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any faster and they will get confused

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so two plucks per second can be[br]thought of as the rate

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or capacity for sending information[br]in this way

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and it turns out that the most[br]common roll is a 7

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so it takes 3.5 seconds[br]to send the number seven

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Alice then realizes[br]they can do much better

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if they change their coding strategy

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she realizes that the odds of each number[br]being sent follow a simple pattern

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there is one way to roll a 2,[br]two ways to roll a 3

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three ways to roll a 4,[br]four ways to roll a 5

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five ways to roll a 6 and six ways to[br]roll a 7, the most common

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and five ways to roll an 8

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four ways for a 9 and so on

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back to one way for a twelve

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and this is the graph showing the[br]number of ways each result can occur

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and the pattern is obvious

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so now let's change the graph to number[br]of plucks versus each symbol

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she proceeds by mapping the most[br]common number, seven

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to the shortest signal one pluck

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she then proceeds to the next[br]most probable number

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and if there is a tie[br]she picks one at random

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in this case she selects[br]six to be two plucks

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and then eight to be three plucks

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and then back to five to be four plucks

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and nine as five plucks

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and back and forth until we reach 12

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which is assigned to 11 plucks

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now the most common number seven

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can be sent in less than a second

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a huge improvement

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this symbol change allows them to send[br]more information

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in the same amount of time on average

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in fact this coding strategy[br]is optimal for this simple example

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in that it impossible for you to come up[br]with a shorter method

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of sending two dice rolls[br]using identical plucks

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however after playing with the[br]wire for some time

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Bob hit on a new idea