How to analyze risk vs. reward for spending on research and development work? Imagine I have a company that makes widgets, where each widget costs me A dollars to make. Each month I can allocate money toward research and development with the aim of finding a new process that will allow me to build widgets for a cost of A/B dollars. Presume that I know that for each C dollars I spend on research and development there's a D% chance of finding a breakthrough. Of course, spending money on research and development means that I have less to spend on building widgets.
I have a monthly budget of E dollars. This budget is not directly tied to my profit margin, but it is safe to say that it my profit margins influence future budgets (i.e., if I make no widgets for three straight months b/c I do all research and development, it's likely that my budget will be reduced, whereas if I discover a breakthrough the first month my profits will skyrocket and I'll likely see that budget grow over time).
In case that is too abstract, here's the real world scenario I'm interested in solving (although I'd like a more general approach, as well):


*

*A = 15 dollars

*B = 3

*C = 5 dollars

*D = 2.75%

*E = 30 dollars


That is, today widgets cost me 15 dollars to build but if I can find a breakthrough I know I can make them at 1/3 the cost (5 dollars). For each 5 dollars I spend on research and development there is a 2.75% chance I'll find the breakthrough. However, I have only 30 dollars to spend each month. If I spend it all on research and development and have no success then I have made no widgets for sale. If I spend it all on widget construction I have no chance of finding a breakthrough.
Is there some statistical distribution or formula that can let me plug in these variables and see some sort of breakdown that gives me an idea of whether it's a good idea to spend any money on research and development each month and, if so, how much?
 A: Since your question is about seeing how your different options play out, and you have a small number of them (the point of my previous question), you can use a decision tree.  (From Wikipedia: "In decision analysis, a 'decision tree'... is used as a visual and analytical decision support tool, where the expected values (or expected utility) of competing alternatives are calculated.")
You will want the first level of branches of the tree to be the various decisions you have (spend 0 dollars on research, spend 5 dollars on research, etc.).  The second level of branches of the tree will be whether or not the breakthrough is found.  Based on the outcomes and probabilities for the second level, you can calculate an expected value for each option in the first level.  Presumably you want to choose the option with the highest expected value (or expected utility, if you are comfortable with utility functions).
However, I think you need some more information to get a really good answer.  In particular, your example doesn't quantify exactly how your budget would change over time.  That needs to be included somehow.  In addition, perhaps going with the highest expected monthly value isn't what you really want, as your description of the scenario implies that you might be willing to go three months without profit in order to increase your chance of a breakthrough.
In short: I think a decision tree is what you want (and there are lots of resources out there besides the Wikipedia page that can help you with that), but you need to quantify some of the other aspects of your problem before you can get a good answer.
A: I know something about your "business" so I'll make a minor correction, and then use a standard ROI analysis.
Your "widgets" will still cost 15 dollars to make if you make your research breakthrough. But they will be "superwidgets" that are three times more productive than the old ones, and therefore worth 45 dollars. So the research will enable you to gain #0 dollars on each widget you make. That should put you way ahead of the competition (unless they develop a competing product), but the odds are somewhat long.
At 5 dollars a pop with a 2.75% chance of success each time, it should cost you 180 dollars to make the discovery on average. That represents sis months of cost, which means that there is a very good chance you won't make it in a meaningful time (one year or less). 
And the return on investment (ROI) is something like $30/$180 or almost 17%. Knowing the nature of your business, there is a good chance that you will fail within a year (probably knowing in about six months), which is to say it is a highly risky undertaking. Such businesses have a required ROI of 25%-30%, meaning that your 17% doesn't "cut it."
There's one other concern I happen to know that the 180 dollars could yield several additional breakthroughs. None of them are quite as impressive as the "superwidget." But let's say, for the sake of argument, that the others ALTOGETHER are worth another 30 dollars. Then your ROI would be more like $60/$180= 33%, which just might "make it."
There are other factors that come into play. That 180 dollars is a guesstimate, and even paying part of it threatens to drain your business. How good is your management relative to your competitors? If yours is better, you might not want to take the chance (unless the competition does). If yours is worse, go for the "gamechanger," because otherwise, you might be competed out of business anyway.
That's all for now. Until we meet again on the other site.
A: This is a decision-theoretic problem. I suggest you look into multi-armed bandits, which

have been used to model the problem of managing research projects in a large organization, like a science foundation or a pharmaceutical company.

If you want a book, there's Multi-armed Bandit Allocation Indices, 2nd Edition by Gittins et al.
A: The other answers are better, but knowing something about the marketplace for your widgets I will add a few more considerations. 
If (1 & A) you are on an island with consumers (or manufacturing bases) in distant lands that are susceptible to hostile takeovers that will dramatically decrease your monthly profits quickly, it is best to take an all or nothing approach to R&D. Either defend your current global positions as much as feasible, or sit back and research until super widgets allow you to retake market share quickly. 
If (2) you are free from hostile takeovers because of the strength of your moat and can all but guarantee a steady source of monthly income for some time, it is more beneficial to conduct steady R&D while increasing your capacity to enter foreign markets quickly when your research bears fruit. 
If (3 & B) you are in an immediate struggle for local control of the market, you may need to mirror the actions of your competitor more closely to avoid a collapse of your position because of an error in judgment. Alternately, it may be beneficial to take advantage of your competitor's spending on R&D to quickly take over their market share and turn their research, even if successful, into a pyrrhic victory. 
In either case, consider the benefits of proper collusion. You and friendly industries need not all research super widgets if you are in agreement on future market share divisions. Spreading the cost of R&D as appropriate may yield the results you seek, while allowing the weaker friendly industries to defend their current market share or make headways in taking over hostile foreign markets. 
