After Antorus raid was released, I started saving the activity feed API data for around 10000 high geared EU characters. From this data I could parse the following statistics about Aman'Thul's Vision and other Pantheon trinket drop rates:
--- Update Apr 7, 2018: Added week 19 data ---
These numbers include only characters who killed Argus the Unmaker every week on normal, HC or mythic difficulty and did not have the trinket before the latest observed week (Error margin means confidence interval of 99% calculated using Agresti-Coull method):
Aman'Thul's Vision:
Week 1: 10448 characters -- 28 drops - Drop rate: 0.270% (error margin 0.162% - 0.437%)
Week 2: 9729 characters --- 14 drops - Drop rate: 0.140% (error margin 0.068% - 0.288%)
Week 3: 9210 characters ---- 6 drops - Drop rate: 0.065% (error margin 0.016% - 0.185%)
Week 4: 8440 characters ---- 8 drops - Drop rate: 0.095% (error margin 0.032% - 0.237%)
Week 5: 7757 characters ---- 5 drops - Drop rate: 0.064% (error margin 0.012% - 0.203%)
Week 6: 7426 characters ---- 5 drops - Drop rate: 0.067% (error margin 0.012% - 0.212%)
Week 7: 7046 characters ---- 7 drops - Drop rate: 0.099% (error margin 0.029% - 0.264%)
Week 8: 5625 characters ---- 6 drops - Drop rate: 0.107% (error margin 0.026% - 0.305%)
Weeks 9-12: 10872 kills ---- 7 drops - Drop rate: 0.064% (error margin 0.019% - 0.171%)
Week 13: 38064 characters - 41 drops - Drop rate: 0.108% (error margin 0.071% - 0.161%) (started to track new chars)
Week 14: 8372 characters -- 11 drops - Drop rate: 0.131% (error margin 0.055% - 0.287%) (all got ilvl1000 trinket on week 13)
Week 15: 8240 characters -- 13 drops - Drop rate: 0.158% (error margin 0.072% - 0.324%)
Week 16: 6560 characters --- 8 drops - Drop rate: 0.122% (error margin 0.041% - 0.304%)
Week 17: 5868 characters --- 8 drops - Drop rate: 0.136% (error margin 0.045% - 0.340%)
Week 18: 5210 characters --- 6 drops - Drop rate: 0.115% (error margin 0.028% - 0.329%)
Week 19: 4596 characters --- 3 drops - Drop rate: 0.065% (error margin 0.001% - 0.278%)
->
TOTAL avg for weeks 2-19: 143015 kills - 148 drops - Drop rate: 0.104% (error margin 0.084% - 0.128%)
Other Pantheon trinket:
Week 1: 10448 characters - 1776 drops - Drop rate: 17.0% (error margin 16.1% - 18.0%)
Week 2: 8134 characters -- 1409 drops - Drop rate: 17.3% (error margin 16.3% - 18.4%)
Week 3: 6345 characters -- 1338 drops - Drop rate: 21.1% (error margin 19.8% - 22.4%)
Week 4: 4598 characters -- 1235 drops - Drop rate: 26.9% (error margin 25.2% - 28.6%)
Week 5: 3068 characters -- 1076 drops - Drop rate: 35.1% (error margin 32.9% - 37.3%)
Week 6: 1919 characters --- 909 drops - Drop rate: 47.4% (error margin 44.4% - 50.3%)
Week 7: 962 characters ---- 581 drops - Drop rate: 60.4% (error margin 56.2% - 64.5%)
Week 8: 293 characters ---- 292 drops - Drop rate: 99.7% (error margin 96.8% - 100.0%?) *see notes
Total amount of characters having a trinket after 8 weeks, from those 5665 who have killed Argus every week:
0.81% has Aman'Thul's Vision after 8 weeks
99.98% has epic Pantheon trinket after 8 weeks (100%?)
Full spreadsheets:
Drops for weeks 1-5
Drops for weeks 6-8
------
Latest notes and conclusions: After having full data for week 8, I can now say that 100% of the 5665 characters on my list have got at least 1 Pantheon trinket. 2 missing drops I verified thru logs site, and for only 1 person I could not verify an epic trinket drop, but he got Aman'thul's already on 2nd week and it is very likely that I'm just missing the part of data where the epic trinket dropped. Based on these results it seems very likely that the drop rate for first trinket is intented to be 100% after 8 weekly kills.
This bad luck protection seem to be based on how many weeks you have killed Argus (one chance each week, LFR not eligible). So on weeks 3-7 your drop rate increases by 30% each week and should be guaranteed drop on week 8 for your first epic Pantheon trinket. For 2nd trinket the drop rate has been much lower, but it probably have some BLP also. Getting exact numbers would be quite complicated but as a Rough estimate I'd guess the off-spec drop rate starts around 5-8% and at some point starts to increase at 15-30% rate weekly. I also tested few fresh characters who just got their first kill on week 8 and their drop rate was below 20%, which correlates with the first week stats above.
If you have not received any Pantheon trinket after 8 weekly Argus kills (or even 9), possible reasons can be: It might be lost in your bags (I've also tested you can't vendor or delete Pantheon trinkets). Or if you had a disconnect during some kill, the game might not count it. Also some other rare looting bug can be possible.
For Aman'thul's Vision the average drop rate has been 0.1% based on 19 weeks of data tracking, with no noticeable changes. Weeks 2-19 have all been within error margin from 0.1% and only week 1 drop rate had statistically significant difference.
--- older notes below ---
Notes for week 7-8: For epic Pantheon trinkets the drop rate has jumped from 60% to near 100% for week 8. Actually I think the drop rate is (or should be) very likely exactly 100% for those who have killed Argus on all 8 weeks. The sample size is limited, but considering that 99.87% of the characters in my data have at least one trinket now, the missing 0.13% is most likely caused by missing data due to the fact that sometimes the activity feed API data is not showing a drop even if it happened.
Notes for week 6-7:
Epic trinket drop rate continues to increase around 30% each week. I have also checked the off spec drop rate for those who can get more than 1 epic trinket, and it's been in 5-10% range each week, possibly slightly increasing but it's clearly lower than the drop rate for first trinket.
Notes for week 5-6:
The drop rate for epic Pantheon trinkets continues to get higher, already getting close to 50% chance, but about 14% of characters still don't have it after 6th Argus kill. Aman'Thul's Vision drop rate still stays very low, unable to have very accurate measurement with this sample size.
Notes for week 4+:
The drop rate for epic Pantheon trinkets is getting noticeably higher each week, which tells that there's some bad luck protection or flat base chance increase for everyone. Aman'Thul's Vision still isn't showing any noticeable increase in drop rate, but it is so low that it can't be measured very accurately with this sample size. I could try to track more characters, although there isn't much practical use for more accurate number as long as the drop rate stays below 0.5%.
For week 5 I've tracked already 5400 kills, and that data is showing epic trinkets drop rate getting already close to 35%, while Aman'Thul is still staying around 0.1%.
Notes for week 3:
For this week the drop rate for epic Pantheon trinkets have increased slightly, but for Amanthul the drop rate is getting even lower. For trying to analyze the accuracy of these numbers, I added confidence intervals using this calculator. Maybe more data is needed, but the difference from first week is already getting statistically significant, which means that the difference is more likely caused by either inaccuracies in data or the actual drop chance in the game has changed. But for now have to wait and see how week 4 is going to be.
Notes for week 2:
The drop rate for Aman'Thul's Vision seem to be around 0.1 - 0.3%. So far there isn't any indication about "bad luck protection", but the sample size is limited and it's been only 2 weeks for now and there's always a possibility that the drop rates will increase in the future. I will keep saving the activity feed data and see later if there's any changes in drop rates. Also what can be confirmed from the timestamps is that Pantheon trinkets drop only from the first Argus kill of the week, regardless of difficulty, and they can't be looted from bonus rolls either. Also I didn't see any noticable difference in drop rates depending whether a character killed first normal or heroic Argus.
General info:
The method used here is similar as I used earlier in my Legendary drop rates thread. Basically I made a list of 10000+ characters from wowprogress, which includes only high ilvl characters from most EU-english servers. Then I'm saving their data from Blizzard's activity feed API few times in a week and have a simple code searching for Pantheon trinket item drops and Argus the Unmaker kill timestamps from this data. Also the equipped trinkets for each character are checked in case the actual drop is missing from the activity feed data(this happens sometimes but not often). Only characters who have killed Argus every week are counted. Then I put this data into spreadsheet for calculating drop rates. For epic Pantheon trinket drop rates I include only characters who did not have any epic Pantheon trinket before, and for Amanthul only characters who did not have Amanthul before.