# A10 Math: The Long Version

Here, we'll give you the gory math details behind the "world according to you" in answer A10. You may wish to read the friendly explanation under A10 first.

Kinds of Disruption
At each year T, a disruption to business-as-usual has happened, or it has not. If:

P8: A disaster has happened before T, or
P9: AI has been created before T, then

A10: Business-as-usual has been disrupted by time T.

So to find A10, we need to take P8 and P9, then combine them. P8 and P9 were calculated in previous answers: A8 and A9, respectively. If you'd like to know how this was done, take a look at the gory math details for A8 and for A9.

Combining Disaster and AI
We have the probability a disaster happened, and the probability AI exists. What we want is the probability that a disaster happened or AI exists. Correcting for double-counting cases where both happen, we have:

Prob(A10: business-as-usual has been disrupted) = Prob(P8: a disaster has happened) + Prob(P9: AI has been created) - Prob(P8 and P9)

Conveniently enough, we're assuming both kinds of disruption are probabilistically independent. So the probability we're looking for in the last term is the product of the individual probabilities:

Prob(A10) = Prob(P8) + Prob(P9) - Prob(P8) * Prob(P9).

That's it! We now have a probability for A10 each year. This is what you see in the graph.

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