Calculation of probability of an event over time when probability is a function of time.

Question

I have a question regarding calculation of an event occurring at least once over some time interval when the probability of the event occurring at any instant is changing continuously during the interval.

The question is related to this one. But in that case the probability of the event is constant.

My assumption is that I need to split the interval up into discrete time steps and then evaluate the classical equation over those time steps.

As an example, suppose I split the interval into N time steps and, for the sake of discussion, suppose the probability increases linearly over the interval from (0.85/N) to 0.85 as shown in this MATLAB style lambda function.:

pp = @(N) (.85/N):(.85/N):.85;

I could then define a vector, P, holding the probabilities for 10 time steps along the interval as

>> P = pp(10)

P =

    0.0850    0.1700    0.2550    0.3400    0.4250    0.5100    0.5950    0.6800    0.7650    0.8500

Then the probability of the event NOT occurring, Pn, for each individual step is given by .

>> Pn = 1-P

Pn =

    0.9150    0.8300    0.7450    0.6600    0.5750    0.4900    0.4050    0.3200    0.2350    0.1500

And the probability the event does not occur over the entire interval, Pi, is given by

% prod calculates the product of the values in the vector here
>> Pni = prod(Pn)

Pni =

   4.8065e-04

Giving the following for the probability that the event occurs at least once in the interval:

>> Pi = 1-Pni

Pi =

    0.9995

All this seems clear enough, but it is incorrect. The problem is that the results are very dependent on the number of intervals I choose and asymptote to 1 as N increases. What I want is the result to either not change as N increases, or possibly to asymptote to some more correct version of the total probability that the event will occur at least once.

It occurred to me that the probability of the event occurring at each time step should be divided by the number of time steps. For example

N=100;
pp = @(N) (.85/N):(.85/N):.85;
Pi = 1 - prod(1-pp(N)/N)

Pi =

    0.3498

But this seems incorrect - the probability of the event occuring at least once over the duration does not seem like it should be less than the probability of the event occurring in one time step along the way.

I would so appreciate some guidance on how to better think about this.