Finding the minimum # of items to be sold to meet a goal, based on likelihood Let's say that I have a project and I need to earn some target to be successful, let's say $50,000

The way I earn is by one time donations at various levels.  1, 5, 7.5 (some discounted item), 10, 15, 20, and 25.

If i say that i want the least number of pledges, then it seems obvious that I would like to maximally sell as much $25 items as possible.  

But in reality, there is some sort of distribution that tells me how likely it is that I'll be able to sell each type of items.  For example, if this were a kickstarter project, typically there are fewer higher tier bids and a lot more lower tier ones.  As to how to obtain this type of information, let's say I have found a model, or I can just draw out the histogram myself based on some reasonable guess about the relative potential of a person choosing a pledge level.  

My question is, how do I figure out, given some likelihood about how people tend to pledge, how to reach this particular target?  Perhaps let's say I assume that the distribution is the same no matter how many people end up bidding.  So I guess I really am trying to find the minimum number of bids necessary, given the liklihood of certain bids actually being sold.  

The reason I'm asking this is because I want to try to quantify in some way the importance of exposure.  The more people see something, there's some likelihood they'll be interested, and even if they are, they're going to pick the tier item that suits them, which is typically low.  I know it might be a bit silly to think about this without regard to the quality of the project itself, but I was wondering if thinking about this might provide some insight on how important it is to do proper marketing.  

I'm also interested in a reverse question.  Let's say I can only reach about 5000 people, based on the fact maybe that similar projects to mine only got an average of that maybe people to back them.  Is there a way to figure out how to shape the distribution to reach the goal in a way that sticks to the goal? Or perhaps to start with my initial distribution, and see how to morph it as little as possible to reach the target?

Thanks
Let's start with an environment in which there is only ever a single pledge level. The question at hand is: how do we choose the dollar amount of that level to maximise revenue (income)?

Observation 1: your probability distribution implies a demand curve.

We can plot the cumulative distribution of willingness to pledge and it will probably look roughly like this (with most of the mass at the bottom, as you speculate):



The $y$ value in this figure tells us the probability that a given individual's willingness to pledge (WTP) is no greater than the dollar value on the $x$ axis.

By turning this upside down, we get the probability that somone is willing to pay at least this much:



This tells us, for example, that there is a rouly 55% chance of a randomly selected individual being willing to pledge at least \$10. If we take 100 people then, on average, 55 of them will pledge. Thus, we can think of the vertical axis as being an indirect way of measuring pledge quantities, corresponding with the pledge level (or price) on the horizontal axis. If we swap the two axes, we therefore get something that looks very much like a textbook demand curve:



Given this demand curve, we can use all of the usual tools from the kind of standard monopoly optimisation problem that you can find in and introductory economics textbook. For example, if the CDF in the first figure is given by function $F$ then$$\Pr(\text{WTP}&gt;x)=1-F(x)$$ so the revenue from setting the pledge level at $x$ when the audience consists of $n$ potential pledgers is $R(x)=[1-F(x)]xn$. Just like a standard monopolist's problem, we can compute the revenue-maximizing pledge level ("price") by setting marginal revenue equal to zero:

$$R'(x)=n[1-F(x)]-xnF'(x)=0.$$

Substituting your empirical distribution for $F$ in this equation and solving for $x$ gives your optimal pledge level.



Now, if we allow for multiple pledge levels then things become much more complicated. Firstly, we need a model of the behaviour of people and why it is that some people might pledge more than others.

If, for example, we are working in a Kickstarter-like environment then the main motive would be to get the goodies that come with the higher pledge tier. But that means we need a model not only of the pledge levels, but what you get in return. A complete treatment of this is beyond the scope of this answer, but a fairly readable summary can be found in "Versioning Information Goods" by Hal Varian. If you want a more detailed, technical treatment then the best place to look is "The Economics of Contracts: A Primer" by Salanie.