So our server had this question posted.
What are the chances of an event X to happen by the year 2025, by 2028 and by 2030?
Everyone but one of the user predicted non-monotonically.
i.e. these were his predictions :
- By 2025 : 40%
- By 2028 : 50%
- By 2030 : 45%
Now this is where our server is divided. We can't seem to agree on whether the predictions here should be monotonic or not. Is it logically incorrect to have the third probability less than second probability? If so, why?
The question is about whether an event occurs by a certain date, or equivalently, by a certain time $t$. One way to mathematically model this is to introduce the random time $T$ that the event occurs at, which is a random real number. Then, the event $\{T\le t\}$ is a subset of the event $\{T \le t'\}$ when $t<t'$, so the probability that $T\le t$ is at most the probability of $T\le t'$.
Monotonicity is one of the fundamental properties of probability, that whenever two events $A$ and $B$ satisfy $A\subset B$, then $\mathrm{Prob}(A)\le \mathrm{Prob}(B)$. A mathematical explanation as to why probability is monotonic ultimately has to defer to the definition of a probability measure (see here, for example), although it can be intuited since "every instance of $A$ is an instance of $B$" plausibly means the probability of $B$ can be no smaller than the probability of $A$.