I've tried searching for this kind of question, but the closest I've found is this, but the math is a bit over my head, and it's also tagged with "normal distribution", which I don't think applies to my question.
I found a few others, but they seem to be the inverse of what I'm trying to figure out, and it doesn't appear to be straight forward to reverse their calculations.
My Question:
Basically, I'm trying to understand how you calculate the probability of an event, when time is the only deciding factor.
Let's say that you're looking at a digital clock on the side of a building which only displays hours & minutes (i.e. no seconds), but the clock is partially obstructed by a tree or utility pole or something. So, at any given moment, you can only see either the minutes or the hours, but never both at the same time.
You start out standing in a location such that you can see the minutes, which read "00". Then, you walk to a new position where you can see the hours, which read "12". If it took you "x" seconds to walk to that new location, what is the probability that the minutes have since changed to "01".
My gut reaction is simply to say x/60. But is that actually true? or is there a better way of understanding this kind of thing.
Thanks.
P.S. Obviously, if this where a riddle, then you could just look at the hours first, and then look at the minutes, and know for sure. But I'm just using this example to provide a foundation for understanding the math.
ere is the data. AUTOBOX identified two counter-acting level shifts (visually obvious at periods 84 and 199) while forming a useful equation. 
incorporating memory , seasonal pulses and one possibly unusual data point at period 262. The Actual/Fit/Forecast is shown here with a 25 hour prediction period 
. The residuals from the model (possibly what you mean by values between the "peaks" ) are shown here 
. The ACF of the residuals suggest randomness reflecting a sufficient model 
as contrasted to the ACF of the original 286 values 
. If you have any detailed questions you can either respond here (preferably) or you can set up a chat room. Detecting when a new value is exceptional is straight forward and would be reported as a possible pulse/exceptional value since the equation can't discern based upon 1 reading whether or not it is a one-time event OR the start of a level shift OR a start of a time trend OR a start of a new seasonal pulse OR a start of a error variance change OR simply an error of measurement.
Best Answer
If you assume that (a) the clock is "correct", in that it changes display to a new minute at the instant that the minute starts (to save having to distinguish clock-time from actual time); and (b) the instant in time that you saw the minute digits was uniformly distributed in $[0,60)$ seconds (which is probably a reasonable model), then $x$ seconds later, the time is uniformly distributed on $[x,60+x)$ seconds after $12:00$.
The probability that the full time display at that point is $12:01$ or later is $\min(x/60,1)$ and the chance that it's still showing $12:00$ - i.e. that the correct time is still in $[12:00, 12:01)$ - is $\max(0,1-\frac{x}{60})$.