After nearly six months of work, the "minions" and I have reached our target of 10,000 breeding attempts of Entbrat + T-Rox → Ghazt with 1 Wishing Torch lit. All of the breeding tries were done using the Windows version of the game, with as many variables as we could think of kept as in our previous work. We got 184 Ghazts out of 10,063 tries, an overall success rate of (1.828 ± 0.134)%.
But wait — there's more. It's not that simple.
People have noticed, from time to time, that there seems to be a brief flurry of successes in breeding Ethereals. The effect is like the "500% breeding chance!" bonus in the mobile-app version of the game, but unnanounced. We seem to have run across something like that early on in this research phase. Looking at the results in groups of about 2,000 tries (the groups were defined by date/time, not by even numbers), we had:
| Group | # Tries | # Successes |
|---|---|---|
| 1 | 2059 | 49 |
| 2 | 1944 | 28 |
| 3 | 2021 | 36 |
| 4 | 1989 | 35 |
| 5 | 2050 | 36 |
We didn't expect to see that much variation. The next step was to investigate if some of the data should be discarded. We split the data, ordered by time, into 20 smaller groups with about the same number of tries in each. Each covered about 10 days, so if the effect was indeed due to a short-term temporary bonus, we would only have to discard one of those groups (unless we were very unlucky and have the bonus time split between two groups). On the other hand, the groups were large enough to have the number of successes in each group be statistically significant. This gave us:
| Group | # Tries | # Successes |
|---|---|---|
| 1 | 504 | 6 |
| 2 | 504 | 11 |
| 3 | 504 | 18 |
| 4 | 503 | 12 |
| 5 | 503 | 11 |
| 6 | 503 | 7 |
| 7 | 503 | 7 |
| 8 | 503 | 8 |
| 9 | 503 | 9 |
| 10 | 503 | 6 |
| 11 | 503 | 10 |
| 12 | 503 | 8 |
| 13 | 503 | 6 |
| 14 | 503 | 14 |
| 15 | 503 | 8 |
| 16 | 503 | 7 |
| 17 | 503 | 9 |
| 18 | 503 | 9 |
| 19 | 503 | 4 |
| 20 | 503 | 14 |
The mean (i.e. average) number of successes per group is 9.2 ± 3.28 .
Several of the groups differ noticeably from that mean, but the third group really stands out: at 18 Ghazts, it's double the mean value, and 2.7 standard deviations (σ) from the mean. (That is, (18-9.2)/3.28=2.7 .) Perhaps that group of trials should be dropped from our data.
One method that's used to determine if some data should be eliminated in this way is Chauvenet's criterion. It starts with the assumption that the set of data follows the "normal" bell-shaped distribution, with most of the values near the mean and some falling farther away. If one has only a few points, one expects that they won't differ from each other by a large amount compared with their standard deviation. If one has many data points, one may expect that some "valid" points will be distributed at some distance and shouldn't be discarded.
To apply this method in this particular case, one looks up that "2.7 standard deviations" in a table or on a graph of prediction interval vs. standard score. Here, the prediction interval value is 99.3% — that is, only 0.7% (0.007) of values would be expected to lie more than 2.7 σ from the mean. Multiply that 0.007 by the number of data groups, 20, to get 0.14 . Chauvenet's criterion says that if that value is less than 0.5 — which it is! — that set should be discarded.
By contrast, we can consider group 19, with 4 successes. (9.2-4)/3.28=1.59 — that is, the 4 value is 1.59 standard deviations from the mean. Looking that up on the chart, it corresponds to a prediction interval value of about 89%; that is, 11% (0.11) of values would lie more than 1.59 σ from the mean. Multiply 0.11 by the number of data groups, 20, to get 2.2. Since that value is much larger than 0.5, we don't discard group 19.
All of the other groups' values are even closer to the mean than that. So the only group we eliminate is group 3. We're left with 166 Ghazts out of 9560 tries, a success rate of (1.736 ± 0.134)%. (In retrospect, I should have asked the "minions" to do another 500 tries to replace the discarded group. Unfortunately, they had already gone on to the next phase of the research, with 4 torches lit, before I did this aspect of the data analysis. We don't want to waste 300 diamonds for each "minion"!) Notice that that relatively small group of questionable results made a rather large difference in the final calculation!
Now we get to the real question: how much effect can be attributed to the Wishing Torch?
We already know that the success rate for this breeding combination without the torch is (1.030 ± 0.093)%. The difference between the two values is 0.706%. As for the uncertainties... well, I'm not a statistician, so I may be wrong about this. (If so, please let me know!) But I think that the variances (the squares of the standard deviations) are additive — that is, since Pwishingtorch = Pwishing torch+base - Pbase, σ2wishingtorch = σ2wishingtorch+base + σ2base. So: the uncertainty is √((0.134%)2 + (0.093%)2), or 0.163%. So for this combination — and I want to emphasize that constraint! — it appears that one Wishing Torch improves the chance of breeding a Ghazt by (0.706 ± 0.163)%.
The relative uncertainty is rather high: 23%! That's an unavoidable consequence of the probability of success being so low to start with. We've been working on this for more than a year, recorded almost 22,000 trials, and still got only a total of 305 successes. Looking on the bright side, our next set of trials, with four torches, should give us much better statistics.
If I were a betting man (which I'm not) and wanted to assume that the probabilities were nice round numbers, I'd guess that a single Wishing Torch gave a ¾% increase to the chance of success. For this combination. We're also looking at the odds of breeding Entbrats from several different combinations of monsters; I'll be reporting on that soon. Once we have good statistics on that, without torches, we'll be trying to find out what effect the torches have on those combinations as well.
Once again, I'd like to thank all of the "minions" for their diligent work.
