Setting the scene: You've been motivating your friends, guildmates, strangers everyday and plowing all the FPs you can into random people's GBs, all in the pursuit of blueprints. You are meticulous. Day after day you motivate and plow, motivate and plow. You are a motivated, plowing machine. You plow so hard your girlfriend can't keep up, and leaves you, but still you motivate and plow on. Eventually, coworkers start to take notice of your unkempt looks and constant plowing at work. One day, HR takes you aside to tell you that the workplace is safe space and your constant plowing is making people uncomfortable... and also... you're not doing your job. But what do they know? They are not motivated and do not understand the thousands of medals you've received for your constant plowing. But alas, they will never understand. You lose your job, but that just means more time to motivate and plow. Days turn to weeks, weeks turn to months. You lose track of how many nights you've spent lying awake in a bed of empty ramen wrappers, fueled by caffeine and an itch that cannot be scratched. Nothing can satisfy. Drugs, women, Jesus, none of it. Your family tries an intervention, but even they know hopelessness when they see it. You just can't... stop... motivating... and plowing... searching... for that one... last... F'ING... BLUEPRINT FOR YOUR ARC. Too close to home? Yes, with the scene set, I introduce you to my theory: FoE intentionally changes the probability of getting a BP for a given square when it is the last square needed to make a BP set for building or leveling a GB, especially when the GB is an Arc. Motive Before I dive into proof, "why would FoE do this" you ask? If you're here (and still reading this rant), you already know. Diamonds = money and money = ramen and the people at FoE LOVE RAMEN. Proof 1. We should expect to need to collect 25 BPs to get at least 1 complete set The problem of collecting all of a set when the probability of collecting one of each of the set is the same at each opportunity is called the "Coupon Collector's Problem". Generally speaking, people use the example of solving for the minimum number of expected rolls it takes to roll all 6 numbers of a fair dice (the answer is 15). The problem can be solved using a geometric distribution formula which can be applied to "how many BPs should we expect to have to collect in order to collect a full set?" There are online calculators that will do this for you. 2. We should expect to need to collect less than 25 additional BPs to get the 2nd set, and less for the 3rd, etc. The problem of figuring out the number of "opportunities" required to collect multiple sets is called the "Double Dixie Cup Problem" ("DDCP"). While it's much more complicated formula, the proof indicates it should take less "opportunities" to complete a set than it did the previous time. So if it takes 25 to get the 1st set, it should take less than 25 more for the 2nd set, and then less again for the 3rd set. Unfortunately, I don't have the math chops to calculate what this means for FoE and I couldn't find any online calculators out there to do it for me. 3. My 8, non-Arc, 1 BP remaining, GBs are representative of #1 and indicate that BPs are indeed applied at random Of the 8, non-Arc GBs I have with only 1 BP remaining for a set, I've collected an average of 28 BPs all time and an average of 1 complete set for each. This is a little freaky, but goes to back up the probability in #1 (should expect 1 set in 25 BPs). This supports that FoE is using random probability for collecting BPs. HOWEVER... 4. I've collected a total of 57 BPs for the Arc while only accumulating 2 full sets. The probability of this scenario is much lower than it should be it BPs were being applied randomly **This gets a little complicated so I'm attaching a screenshot of my calcs** While we can't get to an exact probability using the Double Dixie Cup Problem, we can use a binomial distribution equation to determine the probability of getting "k" amount of BPs in 1 square given "n" amount of total BPs collected. I applied this formula to each of my GBs where I have 1 BP remaining to determine the probability of ending up in that scenario for that specific GB. The probability of getting 0 BPs in a single square of Alcatraz given collecting 11 total Alcatraz BPs is 27% (pretty likely) The probability of collecting 1 or less BPs in a single square of CDM given collecting 32 total CDM BPs is 11.5% (less likely, but still relatively likely). The probability of ending up with exactly 1 BP in a square given collecting 25 total BPs is 21.7% The average probability of where I ended up for each non-Arc GB is 20.0% The probability of my Arc scenario is 4.0% WHY is the probability of my Arc situation so much lower than expected at 4.0% when the rest of the GBs I analyzed are nearly perfectly aligned with the expectation-based probability??? IT'S A CONSPIRACY Misc Assumptions: does not take into account BP trade-ins however, taking this into account would increase "n" and decrease the overall probabilities therefore this is a conservative estimate Assumes all BP sources apply a BP in a truly random way for each individual event (1/9 chance each time) Caveat: sample size of 1 (me). Need more data!