How do I determine confidence intervals around weighted, probilized events?

Question

How would I go about finding out the confidence intervals around a set of distinct bianary occurrences where each occurence has a different associated probability and each occurence is weighted?

To be more specific, we typically determine a milestone budget by assigning reasonable probabilities to each milestone. Each milestone typically usually has a differnt probability of success and a different dollar value. For example if we need to determine the amount of money we need to hold out for 3 milestones, we would do this: $$P(\text{milestone}_1) \text{cost}(\text{milestone}_1)+P(\text{milestone}_2) \text{cost}(\text{milestone}_2)+\dots $$ So it could look something like this: Budget = (10% x $\$$1,000,000) + (50% x $\$$5,000,000) + (20% x $\$$20,000,000) = $\$$6,600,000.

Each milestone is bianary in that it either happens or it doesn't. so for the first milestone, we are saying that there is a 10% chance that we will need to pay $\$$1M dollars and a 90% chance that we will pay nothing.

Obviously, we won't ever actually be paying the probilized amount of $\$$6.6M. How would I go about determining the probability range around where the spend is most likely to fall. In other words, if I wanted to demonstrate where the actual spend might fall with a certainty of 80% how would I go about calculating this? Note that we have more events in my real world problem...

Thanks in advance for any help. I am not a statistician at all so please forgive my ignorance and any lack of clarity.

Answer 1

Basically you would need to make a probability tree with resulting penalty sums in leafs and sum the whole thing up for a criterion of your choice, possibly using some software for more milestones than few (there will be $2^N$ leafs for $N$ milestones). You will be able to build the distribution too.

For a really huge number of milestones, you'd need a Monte Carlo simulation, so basically just simulate the process huge number of times (using random number generator to decide whether to fire or not certain milestone) and then obtain a histogram of outputs as a distribution approximation.

Answer 2

It rather depends on how many milestones you have. But if this (call it $n$) is small enough and each one either happens or does not, then you can work out the $2^n$ possibilities, working out the probabilities by multiplying the inidividual probabilies.

So for example the probability of paying out $20,000,000$ is $0.9 \times 0.5 \times 0.2 = 0.09$ or 9%.

You then sort by value and add up the probabilities, to find for example that the probability of paying $20,000,000$ or less is $0.89$; adding up in reverse you find the probability of paying $20,000,000$ or more is $0.20$.

Your table might look like this

Prob    CumProb RevCum  Amount

0.36    0.36    1.00            0
0.04    0.40    0.64    1,000,000
0.36    0.76    0.60    5,000,000
0.04    0.80    0.24    6,000,000
0.09    0.89    0.20   20,000,000
0.01    0.90    0.11   21,000,000
0.09    0.99    0.10   25,000,000
0.01    1.00    0.01   26,000,000

This will then let you make statements about confidence.