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<h1>A6 Math: The Long Version</h1>
<p>Here, we'll give you the gory math details behind the "world
according to you" in answer A6. You may wish to have read read the
friendly explanation under A6 first.</p><br><br><b>Neuromorphic or Non-Neuromorphic?</b><br>The model assumes AI can come in two forms: neuromorphic or non-neuromorphic. In any year T, if:<br><br>P4: Neuromorphic AI has been created before T, or<br>P5: Non-neuromorphic AI has been created before T, then<br><br>A6: AI has been created before T.<br><br>So to find A6, we need to find P4 and P5, then combine them.<br><br><b>Setting P4</b><br>P4
records whether neuromorphic AI has been created. At each year, it can
say, "yes: neuromorphic AI has been created", or, "no: neuromorphic AI
has not been created". The probabilities for these were calculated
earlier, in A5, based on your inputs. If you'd like to learn how this
was done, <a href="file:///Users/mike/Sites/theuncertainfuture.com/ufHelp/A5Math.html">see the gory math details for A5</a>.<br><br><b>Setting P5</b><br>P5
records whether non-neuromorphic AI has been created. This question we
did not split into different parts: in Q6, you got to choose directly
how probable such AI was at what time. Specifically, you got to set the
parameters <img src="A6Math_files/png_003.png"> and <img src="A6Math_files/png.png"> of a <a href="http://en.wikipedia.org/wiki/lognormal_distribution">lognormal distribution</a>. These represent the mean and standard deviation of the logarithm of the number of years after the starting point. The <a href="http://en.wikipedia.org/wiki/Probability_density_function">probability density</a> that P5 will first become "non-neuromorphic AI has been created" <i>at time t</i> is:<br><br><img src="A6Math_files/png_002.png"><br><br>and the cumulative probability that P3 will have become "non-neuromorphic AI has been created" <i>any time before time t</i> is:<br><br><img src="A6Math_files/png_004.png"><br><br>(If you want to know why we used a lognormal distribution, read <a href="http://www.theuncertainfuture.com/faq.html#lognormal">our FAQ entry</a>. 1950 was chosen as the zero point for the lognormal because it was the approximate beginning of serious AI work.)<br><br><b>Combining Both</b><br>We
have the probability neuromorphic AI exists, and the probability
non-neuromorphic AI exists. What we want is the probability that
neuromorphic AI <i>or</i> non-neuromorphic AI exists. Correcting for double-counting cases where both happen, we have:<br><br>Prob(A6:
AI has been created) = Prob(P4: neuromorphic AI has been created) +
Prob(P5: non-neuromorphic AI has been created) - Prob(P4 <i>and</i> P5)<br><br>Conveniently enough, we're assuming both kinds of AI are <a href="http://en.wikipedia.org/wiki/Independence_%28probability_theory%29">probabilistically independent</a>. So the probability we're looking for in the last term is the product of the individual probabilities:<br><br>Prob(A6) = Prob(P4) + Prob(P5) - Prob(P4) * Prob(P5).<br><br>That's it! We now have a probability for A6 each year. This is what you see in the graph.<br><br><a href="javascript:window.close()">Return to your calculation</a></div></div></body></html>
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