Probability of finding a new smallest value I am doing some simulations and I need a good heuristic for when to stop the simulation. The simulation continuously outputs data with a cost in the range [0,1]. The output looks gamma distributed or possibly Poisson.

This is a standard Google Sheets histogram; all default values.

And here's a scatter plot, with the first attempt at the left side. The orange dots are blue dots which are also the lowest cost seen so far. Those are the ones I need to predict.
What I would like to know is a rough probability of the next simulation yielding a new lowest cost, so that I can do a somewhat informed decision of when to stop. 
This dataset consists of 11,475 values. If you would like the actual data or charts from a bigger dataset or from another run, let me know. This is part of a program I'm writing as a freetime project; it is not part of an assignment.
EDIT: I would guess that all values are independent, since they are executed in parallel. They take the same input, but use different random values.
 A: From the scatterplot it looks like the data are independent.  (There are many ways to test this: consult information on time series analysis and autocorrelations.)  Assuming this is the case, the chance of any value in a set of $n$ values (not just the very last one) being smallest does not depend on when the value occurred.  Consequently all values have an equal chance of being smallest.  Assuming there are no ties for smallest, that chance must be $1/n$ so that all chances sum to unity, as probabilities must.
As an application, you have $11,475$ values.  The next one will bring $n$ up to $11475+1 = 11476$.  Therefore the chance that it is the smallest is $1/11476$.

Incidentally, this rule implies the expected number of new minima among such a sequence is approximately $\log(n)$.  The illustrated data appear to have $9$ new minima (plotted as orange points) and indeed $\log(11475)\approx 9.35$.  A quick simulation (taking ten seconds or so) gives us a sense of the distribution of numbers of new minima, which might be of some interest (and perhaps help inform the intuition).

This is the R code to produce such a simulation.
N <- 1e4
sim <- apply(matrix(runif(11475 * N), ncol=N), 2, function(x) {
  y <- cummin(x)
  n <- length(y)
  sum(y[-1] != y[-n])
})
hist(sim, breaks=seq(min(sim)-1/2, max(sim)+1/2, by=1),
     xlab="# new minima", main="Histogram of Simulated Results")

