How to I extrapolate probability over a time period? The probability of a "success" is 16% in 5 years. What is the probability of success in 10 years? How much time do I need for the probability to reach 70%? Is there a way to answer these questions just by applying math/statistics/... to the data I have got above (a success is 16% likely in 5 years)?

If the arrival of "success" is random with probability $16\%$ in a 5 year interval, then we can fit a Poisson model with rate parameter $\lambda$ to this to determine the Probability of no successes in an arbitrary number of years.

Let $X_L$ be the number of successes in an interval of length $L$:

$P(X_L=n) = Poi(X_L;\lambda) = Poi(X_L;\phi L)$ where $\phi$ is the "normalized rate" (e.g., per year, second, or whatever units we have for $L$).

Then, for a given $L$, we get:

$P(X_L>0)=1-P(X_L=0) = 1-Poi(0;\phi L)=1-e^{-\phi L}$

Since we know $P(X_5=0)=0.84$ this means that $e^{-5\phi}=0.84 \rightarrow \phi=-\frac{\ln 0.84}{5}$

Thus, $P(X_{10}>0)=1-e^{\frac{\ln 0.84}{5} \times 10}=1-0.84^{\frac{10}{5}}=1-0.84^2$ Just as Tunococ derived.

This post just shows how you can derive it from a Poisson model.