Need Statistical Help!

[sambal]

Originally Posted by Estu [You must be logged in to view images. Log in or Register.]

So in the previous case, each one has a 50% chance to drop and 75% of the time we get at least one bone chip, but the expected number of bone chips is 0*25% + 1*50% + 2*25% = 1, so the wiki would show a "likelihood" of 100%. These cases would, I'm assuming, complicate the confidence intervals, though most pieces of loot that drop can only drop once, so it's more straightforward.

Any stats nerds wanna help me out? Thanks!

I don't think you're 100% correct here.

The skeleton is drawing from the possibility of dropping two bones, each with the probability of 50%. This is the way almost every video game works, which does wind up yielding a normal distribution. It's much easier for the server to compute these straightforward loot tables, and appear to be convoluted to the player.

p_Bone_A = 0.5
p_Bone_B = 0.5

Probability that bone_A does not drop, P(A'):........................... 0.5 (no bone_A, irrespective of bone_B)
Probability that bone_B does not drop, P(B'):........................... 0.5 (no bone_B, irrespective of bone_A)
Probability that bone_A or bone_B drops, but not both:............. 0.5 (one bone)
Probability that bone_A and bone_B both drops, P(A∩B):............ 0.25 (two bones)
Probability that bone_A and/or bone_B drops, P(A∪B):............... 0.75 (one or two bones)
Probability that neither bone_A or bone_B drops, P(A'∩B'):.......... 0.25 (zero bones)

[sambal]

p_bone = 1 works because over enough time, the player will average out having 1 bone for every skeleton killed. (for observations over about 30)

[Estu]

Originally Posted by sambal [You must be logged in to view images. Log in or Register.]

I don't think you're 100% correct here.

The skeleton is drawing from the possibility of dropping two bones, each with the probability of 50%. This is the way almost every video game works, which does wind up yielding a normal distribution. It's much easier for the server to compute these straightforward loot tables, and appear to be convoluted to the player.

p_Bone_A = 0.5
p_Bone_B = 0.5

Probability that bone_A does not drop, P(A'):........................... 0.5 (no bone_A, irrespective of bone_B)
Probability that bone_B does not drop, P(B'):........................... 0.5 (no bone_B, irrespective of bone_A)
Probability that bone_A or bone_B drops, but not both:............. 0.5 (one bone)
Probability that bone_A and bone_B both drops, P(A∩B):............ 0.25 (two bones)
Probability that bone_A and/or bone_B drops, P(A∪B):............... 0.75 (one or two bones)
Probability that neither bone_A or bone_B drops, P(A'∩B'):.......... 0.25 (zero bones)

How is this different from what I said?

[sambal]

Originally Posted by Estu [You must be logged in to view images. Log in or Register.]

How is this different from what I said?

I am saying don't focus on the probability of getting 0, 1 or 2. It doesn't really make sense. You don't even need to focus on the probability of each event happening independently of each other.

However, you do need to focus on the probability of getting one bone chip. That's 1. The underlying mechanism doesn't really matter to the player. Why bring it up?

On top of all that, if you really have 100 observations, observed loot will be very close to the actual loot tables.

[seped]

Bugged my statistician friend, he pointed out an item from the page I linked earlier http://en.wikipedia.org/wiki/Binomia...score_interval That is specifically designed for smaller sample sizes or extremes of probability. That might be just what you want.

[Estu]

Originally Posted by seped [You must be logged in to view images. Log in or Register.]

Bugged my statistician friend, he pointed out an item from the page I linked earlier http://en.wikipedia.org/wiki/Binomia...score_interval That is specifically designed for smaller sample sizes or extremes of probability. That might be just what you want.

Right, I noticed that earlier and wrote in this thread that I'd try it. Thanks again!

[koros]

This is my shit right here. I'll write up a post this weekend on the best way to do this if I get time this weekend.

[Estu]

Originally Posted by koros [You must be logged in to view images. Log in or Register.]

This is my shit right here. I'll write up a post this weekend on the best way to do this if I get time this weekend.

Thanks, but I've already coded up a method using the Wilson score interval, and it looks pretty good. It'll be in the wiki pending approval from Ravhin. I'd still welcome more discussion but I'm enjoying the Wilson interval at the moment.

[Splorf22]

So here is something to ponder Estu. Take your Dwarf Skeleton example. The Wilson test will say 1.5-2.7% probability for the Rusty Bastard Sword, and 0.1-0.5% probability for the Cloth Cap, or something like that. On the other hand, I look at that table and I'm guessing that 50% of skeletons drop a random rusty weapon or cloth dagger, i.e. that all of the items have the same drop probability. Can you really see Nilbog sitting there saying ho ho, 2.7% for the Rusty Bastard Sword and 0.2% for the Cloth cap?

I'm guessing that a Bayesian approach will do well here for stuff that drops many different possible items.

[Estu]

Originally Posted by Splorf22 [You must be logged in to view images. Log in or Register.]

So here is something to ponder Estu. Take your Dwarf Skeleton example. The Wilson test will say 1.5-2.7% probability for the Rusty Bastard Sword, and 0.1-0.5% probability for the Cloth Cap, or something like that. On the other hand, I look at that table and I'm guessing that 50% of skeletons drop a random rusty weapon or cloth dagger, i.e. that all of the items have the same drop probability. Can you really see Nilbog sitting there saying ho ho, 2.7% for the Rusty Bastard Sword and 0.2% for the Cloth cap?

I'm guessing that a Bayesian approach will do well here for stuff that drops many different possible items.

I assume you're talking about 'a decaying dwarf skeleton' based on the numbers you're giving. I've run it through my program, and here is what it says for those two items:

Number of decaying dwarf skeletons killed: 415
Proportion of them holding rusty bastard swords: 2.2%
Proportion of them holding small cloth caps: 0.2%

95% Wilson confidence interval for rusty bastard swords: 1.1%-4.1%
95% Wilson confidence interval for small cloth caps: 0%-1.4%

A user looking at this could conclude that there is a 1% drop rate for all the 'common' items (that 1.1% lower bound on the rusty bastard swords is pretty close to 1%, and given that there are about 20 common items, it's not surprising that one or two would fall outside of the 95% confidence interval). My conclusion is that the Wilson confidence intervals give good results that are consistent with reasonable assumptions (i.e. that all these items in reality drop at the same rate). However, I don't know the theoretical or practical differences between Wilson intervals and a Bayesian approach, so I'd be interested in hearing about it.