Pop rates

For the last couple of weeks I've been working on equipping some of my underdressed character I use less. Part of this has been something that most people have done plenty of times - killing Justice in the Tower of Enlightenment. I usually kill Justice using just one of my thieves. It takes around 100 heals and a couple of true sights per kill. Fairly straight forward just need to watch the circle lag. Eventually the result is this:

Your pierce tears Justice!
Justice is DEAD!!
As the final blow is struck, a necklace of shimmering scales falls from
Justice's neck.
Justice splatters blood on your armor.
You get a gold coin from the corpse of Justice.
The corpse of Justice holds:
     (Humming) the Blindfold of Clarity
     (Glowing) (Humming) sword of equilibrium
     (Glowing) (Humming) scales of alpha & omega

The big mystery is once you've killed Justice what alignment the scales of alpha and omega are. Good aligned ones are the most useful and the most rare. In my experience I find that out of every 6 kills I get 3 neutral align, 2 evil and 1 good. In the last two weeks I've done 10 kills resulting in 6 neutral, 2 evil and 2 good aligned scales. I was wondering how likely this result was so started thinking about killing Justice as a Bernoulli trial. In a Bernoulli trial we have an event which has probability of occurring and can be repeated. In this case the event is getting a good aligned scale or not and can be repeated as in each kill. Looking at an individual binomial distribution table using a probability of 0.15 (roughly 1 in 6) for each kill the number of good align scales being popped in 10 kills is:

• 0 good scale - 0.272
• 1 good scale - 0.385
• 2 good scales - 0.238
• 3 good scales - 0.084
• 4 good scales - 0.018
• 5 good scales - 0.003
• 6 good scales - 0.000

Looking at the above getting 2 good scales out of 10 kills is quite lucky - but this is making the assumption that the pop rates are 1 in 6 each kill.

Earlier this year I donated 20 Unfettered Deliriums to the Guild of Vampires. The aim was to see how many Maniacal Tendencies we could make from them and use them as prizes for guild quests. The pop rate for them is lower than for good aligned scales. In the end we managed to get 2 MT's out of 20. My guess for the pop rate from experience would be either 1 in 20 or 1 in 10. Again using an individual binomial distribution table to guess how many Maniacal Tendencies we should have popped gives different probabilities.

Using 1 in 20 gives:

• 0 MT's - 0.358
• 1 MT's - 0.377
• 2 MT's - 0.189
• 3 MT's - 0.060
• 4 MT's - 0.013
• 5 MT's - 0.002

While 1 in 10 gives:

• 0 MT's - 0.122
• 1 MT's - 0.270
• 2 MT's - 0.285
• 3 MT's - 0.190
• 4 MT's - 0.090
• 5 MT's - 0.032

The question is which is more likely to result in popping 2 MT's out of 20 attempts? Hard to tell - might need to try and make some more.