If something has a $12\%$ probability of occurring every decade, what is the probability that it will occur in $100$ years? If something has a $12\%$ probability of occurring every $10$ years, what is the probability that it will occur in $100$ years? And also in $150$ years?  

Is the formula for this as simple as $1-{0.88}^{10}$ or is it more complicated?
One possible approach is to regard such rare events as Poisson.

If the Poisson rate is $\lambda_{10}$ for a ten year period and $X_{10}$ is the number
of events that occur in ten years, then we have 
$$P(X_{10} \ge 1) = 1 - P(X_{10} = 0) = 1 - e^{-\lambda_{10}},$$
so that $e^{-\lambda_{10}} = 0.88$ and $\lambda_{10} = -\log_e (.88) =  0.1278.$

Then the rate for 100 years is $\lambda_{100} = 1.278.$ If $X_{100}$
is the number of events in 100 years, then 
$P(X_{100} = 0) = e^{-1.278} = 0.2786$ and $P(X_{100} \ge 1) = 0.7214.$

Of course, this is about the same as your $1 - .88^{10} = 0.7215,$  but you might find it interesting to think about the Poisson approach.
1 - 0.12^10 * 100 = 100 - 6.1917364E-8 %

Try branching out the possibilities. in each branch multiply the percentage to get the odds of that branch happening.