Path of Exile

Bernoulli Trial, Cumulative Distribution Function, and the Trial master

Yesterday Bex posted that it would take on average ~50 T14+ maps to see the Trialmaster (link: https://www.reddit.com/r/pathofexile/comments/n4ahe0/3141b_patch_notes/gwuktc9?utm_source=share&utm_medium=web2x&context=3). Followed by the line: " We've tested this multiple times to make sure this rate is matching the number of encounters actually being done and it does match." I have 0 reason to doubt they are seeing this in their data….but based on how the probability is encoded and how it is being tested their could be a huge discrepancy as to whether a player picked at random feel like they are experiencing the trial master as frequently as once in 50 ultimatums.

The lazy exile's approach:
-Set the trial master to spawn in 1/50 encounters (2%).
-Across all player record the number of T14 ultimatum trials attempted (for easy of math 100 million)
-Across all T14 ultimatum trials record the number of trial masters that pop-up (for easy math 2 million)
-Lazy stats person sees in fact 1/50 ultimatums lead to trial master so concludes players see trial master every 50 maps give or take a few.

The exile with one college course in stats:
-A single ultimatum is a Bernoulli's trial (ie weighted coin flip) with a Success/Failure chance of seeing the trial master. The success probability is 2% (1/50)
-For a given exile the chance of seeing the trial master is the Cumulative Distribution Function which is dependent on the number of ultimatums performed and the probability of success in each trial. This means:
–50% of players will have at-least one trial master within 35 T14+ maps. (Sounds pretty good!)
–20% of players (a whole 1 in 5 of us) will have at least one trial master in 80 T14+ maps (Not so good…80 maps is a long time)
–10% of players (1 in 10…enough to easily drown out a reddit forum) will have at least one trial master in 114 T14+ maps (114-T14+ map is seriously 50% of my entire mapping for a league)
–5% of unlucky players wont see the trial master even after 148 T14+ maps. Sucks to be you exile.
–1% of the player base won't see the trial mater in 228 – T14+ maps. (I loved last league and I didn't get 228 maven witnesses).

Read more:  Unlocking the average POE player using challenge completion data. What makes you the 1% and what is a casual?

Anyway. For 90% of the player base to see the trial master at-least once after 50 ultimatums the probability of encountering him would need to be ~4.6% per ultimatum. Which is close to 1/21 as compared to 1/50 which I fear the rate of success is set to. #neverSeenThatTrialMaster

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