I was thinking of making a top-level post but this seems like a good place for this.
### motive/contextThere was a recent discussion in global chat about the relative merits of artifact digs, including a link to this fine spreadsheet:
https://docs.google.com/spreadsheets/d/1F5ejYV2bYeaP6yTH683A27iTY8ulDFxGXjwh0zXB82s/edit#gid=502188169The figures presented in this spreadsheet suggest that the three poor-or-epic digs are inferior to the eight hour common dig in terms of expected epics per 1000 hours.
However, this is based on a presumption that the numbers displayed in the UI accurately represent the probabilities. If, on the other hand, these probabilities are rounded to the nearest percentage point, then the digs may be equivalent after all, depending upon the actual probabilities.
### dig metricI will evaluate digs in units of "poors per hour", representing how many poor artifacts we expect per hour of digging at any given dig site. Non-poor artifacts are evaluated as such:
* 1 common = 5 poor
* 1 uncommon = 25 poor
* 1 rare = 125 poor
* 1 epic = 625 poor
### two examplesLet us first consider one popular dig, with an eight hour duration and a 100% chance of yielding a common artifact. We expect one common every eight hours, which converts to 5/8 = 0.625 poors per hour.
Next let us consider another example, with a 24 hour duration and a 50/50 chance of either a common or an uncommon artifact. Those 24 hour periods are each expected to yield an average of (0.5 * 5 + 0.5 * 25) = 15 poors, which scales to 0.625 poors per hour again.
### poor-or-epic digsThere are three known dig site classes which yield either poor or epic artifacts. The displayed times and probabilities are:
* 12 hours, 99% poor, 1% epic
* 22 hours, 98% poor, 2% epic
* 32 hours, 97% poor, 3% epic
If we assume these probabilities are accurate, then these digs turn out to be a little bit worse than the eight-hour 100%-common dig, specifically:
* 12 hour dig: (0.99 + 0.01 * 625) / 12 = ~0.6033 poors-per-hour
* 22 hour dig: (0.98 + 0.02 * 625) / 22 = ~0.6127 poors-per-hour
* 32 hour dig: (0.97 + 0.03 * 625) / 32 = ~0.6163 poors-per-hour
However, if these probabilities are simply the result of rounding to the nearest percentage point, then the digs may be equivalent. So which probabilities would make these digs equivalent?
Let's declare some variables:
* p_12 = probability of a poor in the 12-hour dig
* p_22 = probability of a poor in the 22-hour dig
* p_32 = probability of a poor in the 32-hour dig
In that case the three dig sites yield an expected:
* 12-hour: [p_12 + (1-p_12)*625] / 12 = (625 - 624*p_12) / 12 pph
* 22-hour: [p_12 + (1-p_12)*625] / 12 = (625 - 624*p_22) / 22 pph
* 32-hour: [p_12 + (1-p_12)*625] / 12 = (625 - 624*p_32) / 32 pph
If we set each of these equal to 0.625 and solve for p, we get:
* p_12 = (625 - 0.625 * 12) / 624 = ~98.9583%
* p_22 = (625 - 0.625 * 22) / 624 = ~97.9567%
* p_32 = (625 - 0.625 * 32) / 624 = ~96.9551%
Which are pretty close to the displayed values of 99%, 98%, and 97% respectively.
### conclusionI don't know what the actual dig probabilities are, but if the displayed values are the subject of rounding, then it may very well be that all digs are equivalent in terms of expected poor-artifacts-per-hour units.