#### purplelocust

##### Diabloii.Net Member

**Holy Shield on a scepter probability?**

I tried to estimate the probability of generating a scepter with Holy Shield spawning on it, but I'm no expert and am wondering if the following computation is in the right range or if I'm way off:

My understanding is that the only way that a scepter can spawn with Holy Shield is for it to be rolled with skills that cannot spawn on a scepter normally (Holy Shield and Smite) six times in a row, and then a seventh time it must be Holy Shield at which point the algorithm gives up and assigns Holy Shield anyway.

So at the start, when a scepter spawns, there is a 30% chance of no skills, 40% chance of 1 skill, 20% chance of 2 skills and 10% chance of 3 skills.

So let's start with the 3 skills case, which would have the greatest chance of spawning Holy Shield. I am assuming that once 3 skills is selected, the algorithm keeps trying to allocate the 3 skills until successful- if not, the probability would be significantly less.

Suppose a scepter with ilvl greater than 36 is dropped/shopped, giving the 5th tier of skills to chose from. There is a 50% total chance that the skill tier could change- go up to 6th or down to 4th or 3rd. The chance of getting the first tier skill Smite on such a scepter is 0 (which also causes the reroll) so we can ignore that. The five 5th tier skills are Conversion, Holy Shield, Holy Shock, Sanctuary and Meditation. In order to get Holy Shield to spawn, it must be chosen each of the first 7 times.

For each skill choice, the chance of getting HS is:

50% chance of staying in the 5th tier * 20% chance of choosing Holy Shield amoung the 5th tier skills = 10%

So for 3 skill choices, the chance of getting HS at least once is 10% + (90%*10%) + (81%*10%)= 27.1%.

So the probability of getting HS seven times in a row is .271^7=.000107346

And that is assuming we are amoung the 10% of the scepters that get 3 staffmods. For two staffmods and one staffmod, the chance of getting HS seven times in a row is considerably less:

for two: (10%+(90%*10%))^7= .00000893872

for one: 10%^7 = 10^-7 = .0000001

Totalling, the expected values we get

10% * .000107346 +20% * .00000893872 +40% * 10^-7

which gives the probability of 0.0000125623, about 1 in 80,000. That seems pretty small, but since you can generate scepters by shopping and can look at a bunch for free, it doesn't seem as incredibly tiny as I would have thought. I don't know how long it would take to inspect 80,000 scepters in the shops, but that doesn't seem as hard as doing 10,000 Pindleruns and many people apparently have the patience for that!

For other ilvls less than 37, there is also a chance of HS spawning but those chances would be much lower. For example, a scepter ilvl between 24 and 36 would start with tier 4 staffmods with a 20% chance to boost the tier by one to tier 5, so the chance of getting HS on such a scepter would be 4% for each roll instead of 10%. So when iterated seven times, this is far less likely.