For small probabilities, the weighted average calculation is dominated by the high-probability possibilities—if your 50% confidence interval was up to 1 in 10,000, then 25% of the probability probability mass is to the right of 1 in 10,000, so you can’t say anything less than (0.75)x0 + (0.25)x1 in 10000 = 1 in 40,000.
I wasn’t using a normal distribution in my original formulation, though: the mean of the picture in my head was around 1 in a million with a longer tail to the right (towards 100%) and a shorter tail to the left (towards 0%) (on a log scale?). It could be that I was doing something stupid by making one tail longer than the other?
It would only be suspicious if your resulting probability were a sum of very many independent, similarly probable alternatives (such sums do look normal even if the individual alternatives aren’t).
For small probabilities, the weighted average calculation is dominated by the high-probability possibilities—if your 50% confidence interval was up to 1 in 10,000, then 25% of the probability probability mass is to the right of 1 in 10,000, so you can’t say anything less than (0.75)x0 + (0.25)x1 in 10000 = 1 in 40,000.
I wasn’t using a normal distribution in my original formulation, though: the mean of the picture in my head was around 1 in a million with a longer tail to the right (towards 100%) and a shorter tail to the left (towards 0%) (on a log scale?). It could be that I was doing something stupid by making one tail longer than the other?
It would only be suspicious if your resulting probability were a sum of very many independent, similarly probable alternatives (such sums do look normal even if the individual alternatives aren’t).