How to I extrapolate probability over a time period?

Question

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)?

Answer 1

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.

Answer 2

If you can't assume independence from one time period to another, I don't think you have enough information to solve the problem. If you can assume independence, and if you assume that a success can occur at any point in time, then it seems to me that the random variable $X=$ length of time until success, should be exponentially distributed. That is the probability density function of $X$ should be of the form $f(x)=\frac{1}{a}\cdot e^{-ax}$ for $x \ge 0$ (and $0$ otherwise) with $a$ a positive constant that may be determined by the given information of $P(\text{A success within 5 years})=.16$.

From this you would then be able to answer questions regarding the given situation.

Answer 3

It depends greatly on what kind of process you're observing.

Some processes have a very nice mathematical structure such that the chance of "success" in any given 5-year period, given that there was no "success" before that, is the same.

On the other hand, let $X(t)$ be the event that Mary's son, who is now about to start his junior year in college, will graduate from that same college within $t$ years from now. He is a very poor student, so we estimate $X(5)$ at only $16$%. But if he cannot finish two years' remaining studies in $5$ years, what do you think the chances is that he will finish it in the next $5$ years? Consider the likelihood that by that time he will have given up.

The probability that he will ever graduate from that college would appear to be less than $70$%, so no amount of time will suffice to yield such a high probability of success.