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Message: <blockquote data-quote="KarinsDad" data-source="post: 4695640" data-attributes="member: 2011">I had a lengthy response, but ENWorld went down and lost it when I hit submit last night.So, I will give you the Reader's Digest Condensed version.First, I am not solving the former equation. Some other people here might want to solve for that, but I am not. I am solving for how well Shield and Second Chance protect.Having said that, let's take a slightly different look at it. Let's have a 50% chance on average that the Wizard will get attacked on any given round for 16 rounds, a 50% chance that the attack will hit, and a 75% chance that any given attack is versus AC or Reflex.The unconditional percentage = 7.5%n=16 rounds, Prob (Shield works)= 1-(1-0.075)^16 = 71.2745% of the time.Didn't you state that the chance was 72.75%? Obviously, the number of rounds are important if there is a 50% chance to even attack the Wizard. How could you have been mistaken? <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f609.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=";)" title="Wink    ;)"  data-smilie="2"data-shortname=";)" />The reason your first equation is incorrect is the same reason that the third equation here is incorrect. And yes, I absolutely understand that your first equation is the default equation anyone would consider writing when doing this type of problem using normal probability (and why APC is absolutely convinced I am wrong on this, we are comfortable with what is familiar).The number of rounds do not matter. The misses do not matter. The only thing that matters is the hits when calculating this. Rounds (or attacks) which do not involve a hit do not do damage and are not a consideration when figuring out the math. They are non-events. We are only concerned with how well Shield protects. Our set is not a superset of everything, it is a set of when damage occurs.But, I am willing to admit that I make mistakes late at night. If you can explain why your first equation must be correct, but my third equation is incorrect, I am willing to listen.Think carefully about it. While you are at it, make sure that the 20% (or 40%) can be multiplied by the 75% inside the equation. Just because it works for n=1 does not necessarily mean that it is correct.For 4 hits and 4 misses in an encounter, the equation is:1-((1-0.30)^4 * (1-0)^4))which reduces to your second equation.For 4 hits and 10 misses in an encounter, the equation is:1-((1-0.30)^4 * (1-0)^10))which also reduces to your second equation.Shield protects exactly the same in these two encounters, even though the number of attacks are different. Food for thought. I also used "only when hit" when calculating Second Chance. So, since you claim that I upped the odds slightly for Shield, I also must have done that for Second Chance.You started your word problem here with "Suppose that you are hit with an average of 4 successful attacks in an encounter". That is not the word problem your first equation solves.Your first equation solves the word problem: "Suppose that you are attacked with 8 attacks in an encounter".The problem sets are slightly different.</blockquote>

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