I'm trying to figure out how to convert an annual failure rate between periods. Assume failures are uniform and independent.
I know that the quick, back-of-the-envelope way is simply to divide the annual failure rate (let's call it $a$) by the number of periods in a year (call it $p$) to get the failure rate for that period (call it $a_p$). However, that's not exactly correct.
No matter how I subdivide the year, I should get the same failure rate for a 1 year period. My intuition says that the likelihood of one failure in a year is therefore equal to the likelihood of a failure in the first period, or the likelihood of no failure in the first period and a failure in the second period, or the likelihood of no failures in the first two periods and a failure in the third period, etc.
$$a = \sum_{k=0}^{p-1}a_p(1-a_p)^k=1-(1-a_p)^p$$
Is my intuition on this correct?
The quick answer is the right one (divide the total by the number of periods). You should expect deviations from the mean. The relative size of the deviations gets smaller as the number of samples increases. This latter fact is the law of large numbers. But the law does not say that the sample mean ever necessarily equals the expectation for the distribution.
It's difficult to discuss where intuition fails, but it looks to me that you are either confusing the distribution of samples with the model, or are confusing the discrete and continuous uniform distributions. The uniform continuous distribution has the property that proportional subintervals have the same "rate".
Uniform distribution, Expected value and standard deviation for proportion of observations in a subintervall