How to minimize experience required by crafting items?

Question

Crafting items in sequence requires you to spend experience. The more experience you spend, the more likely you are to succeed. However, the experience required grows very fast as the percentage chance approaches 95% (the maximum).

Here's, e.g., the values for the super-expensive Half Prism at difficulty level 35:

Success%     XP  Success%     XP  Success%     XP  Success%     XP 
---------------  ---------------  ---------------  ---------------
    0.1     289      10      474      40    1,170     89.3   7,499
    1       299      15      574      50    1,505 
    2       318      20      678      60    1,966  I can't go any
    3       337      25      787      70    2,684  higher for lack
    5       376      30      904      80    4,050  of experience.

What's the best strategy then?

Answer 1

*ahem*

The number of tries required to get the first success in a sequence of Bernoulli trials is modeled by the geometric distribution, and its expected value is one over the success probability (e.g., the expected number of tries for a 50% success probability is, unshockingly enough, 1/0.5 = 2).

For difficulty level 35 I get the following:

Success%   Cost  Success%   Cost  Success%   Cost  Success%   Cost
---------------  ---------------  ---------------  ---------------
    0.1 289,000      10    4,740      40    2,925     89.3   8,398
    1    29,900      15    3,827      50    3,010 
    2    15,900      20    3,435      60    3,277  "Cost" is the
    3    11,233      25    3,148      70    3,834  expected value
    5     7,520      30    3,013      80    5,063  of XP spend reqd

Focusing on the interesting bit of cost range seems to hint at 39.0% precisely as the percentage value that minimizes the expected cost. Here's the obligatory misleading graph:

Dots are data. The line is Excel's "smoothed line" interpolation thereof. The big dot is the position of the approximated global minimum.

Unfortunately, I originally assumed things would scale between levels keeping this generic shape. Turns out that's not quite it. On difficulty level 7, for example, (Inhibitor the Second), the sweet spot is 61.1% (chart). At level 19 (key the fourth), instead, the best option would be 47.9% (chart).

As you can see, though, the curves are basically flat at around 50% chance. This means that, if you're not feeling like running the numbers, the default is a reasonable choice (even though not the optimum).

This is my analysis. You can probably do much better than this, for example considering cumulative distribution functions to give confidence intervals or whatnot other statistical goodies I can't think of right now. I'll be giving it a try (I actually already have done so, accidentally... gotta be careful when the game is alt-tabbed away.)