Math geeks, help me out with this one?

See if I'm looking at this right:

I'm trying to collect all the WvW siege decorations for my guild. They are only available via RNG from the WvW Siege Decoration Case skirmish chest reward. There are 21 possible drops from this chest. Obviously we can't say for sure what the drop rates are, but it seems reasonable to assume they are equal, therefore it's roughly a 4.8% chance to get any one particular piece each time you open a chest (trial).

I'm currently missing only one piece - the Guild Ballista. I have opened 177 chests. If my calculation is correct, the probability of getting at least one success (the Guild Ballista) over 177 trials is 99.983452215%, and the inverse, the probability of NOT getting one over 177 trials, is 0.016548%. Makes my situation seems ridiculously unlucky. Is my math right here? Is it just bad RNG or is this item inexplicably rarer than the others? Also why is this box RNG at all?

The maths are correct, assuming an even probability distribution for all 21 drops. Is it ridiculously unlucky, though? Maybeeee. If we assume an active player base of just 100,000 people, there are 16 others just as unlucky as you. With a million players, there would be 165 of you poor sods.

Why not add your data to the Wiki?https://wiki.guildwars2.com/wiki/WvW_Siege_Decoration_Case/Drop_rate

1−((20÷21)^177)=0.999822379100=99.9822379100%Your numbers aren't off but it is also probability.

That's a "lot" of chests but not "a lot" of chests for multiple players (ie a guild) and what ANet seems to feel is appropriate for WvW (15-30 hours a week to cap Pips!).

This would be the difference between theoretical probability and practical probability. In theory, yes, you have your 99.98% chance however in reality those boxes are not linked and thus each and every one of them has the presumed 4.2% of giving you that item, regardless of how many of them you open. It is also entirely possible that the odds are not evenly split. That is a thing games do to entice you to keep getting the box, they give a single item ridiculously low chance of being acquired for no other reason than to prolong the grind/get you to spend more money(depending on the box).

My stats are rusty, so rusty... however, you need to consider these as independent events. Think of flipping a coin... 50% chance for heads and the same for tails. You could do that 100's or 1000's of times, and the chance of getting tails is still 50% for EACH event. Obviously, the statistical probability of flipping it 10 times and not getting a single heads is pretty low (in line with the probabilities that have been presented above).

Essentially, each chest you opened had a 4.8% chance of containing what you were looking for. You're rolling a 20 sided die and looking for a crit (NERD ALERT!), and your luck this session isn't that great. You probably hit 19 a few times... sure the results weren't bad, but where's that 20?!

I hope you crit soon.

@"Kunzaito.8169" said:Obviously we can't say for sure what the drop rates are, but it seems reasonable to assume they are equal, therefore it's roughly a 4.8% chance to get any one particular piece each time you open a chest (trial).

I think this assumption is not reasonable, especially since you didn't get the Guild Ballista after 177 rolls. It looks like there are three categories, Guild Siege, Normal Siege and Superior Siege. My guess is that the items of those categories have different drop rates. Maybe the Superior Siege items have a 10% drop rate, the Normal ones 5% and the Guild ones 1%. In this case, it would be pretty predictable that you don't get all Superior ones after 177 rolls.

@TWMagimay.9057 and @crashburntoo.7431 There is no Gambler's Fallacy in the calculation shown. The math is correct, and it's very unlikely to not get the Guild Ballista when opening 177 chests if the drop rate really was 4.8%. That is why my conclusion is that the items don't have an equal chance to drop.

Did you open 177 chests trying to get the Guild Ballista or trying to get a full set?

If it's the latter there would be 20 others that each have that same chance of not being obtained in that many chests.The chance you don't get a complete set in 177 chests (still assuming equal drop chances) would be 1-((1-((20/21)^177))^21)=0.0037 or 0.37%

In addition to this the wiki drop research page (https://wiki.guildwars2.com/wiki/WvW_Siege_Decoration_Case/Drop_rate) suggests a lower drop rate than 1/21 for the guild siege.

Using 2.5% (a high estimate based on the drop research, assuming all guild siege does have equal odds) as a drop rate the chance becomes:

(1-0.025)^177=0.0113 or 1.13% of not obtaining the guild balista and1-((1-((1-0.025)^177))^5)=0.0553 or 5.53% of not obtaining all 5 guild ones, with additional (smaller) chances of missing any of the non-guild ones.

If my math is right of course (it's been a while since I've done this sort of calculations).

Your math is incorrect. Or, rather, the math is correct, but your statistical analysis is incorrect.

To count all 177, you have to measure the odds for each of the 21 options, from the start. But, you're only looking at one of the 21 items available.

What you really want is the expected wait time RANGE for each of the 21 items. I forget the formula (hey, I haven't seen it in over 40 years, and that book was lost in a basement flood a couple decades ago). But, once you have tabulated your 177 entries, your "wait" starts over. In other words, RIGHT NOW, there is a 1 chance in 21 for each time you check (assuming they are all equal odds). That translates into an expected wait time for THIS item of 21 more turns, from right now.

A probability is, what its meaning tells without ambiguity: A probability = Nothing certain. Whatever the result of the calculation is, it does not tell when you will get what you are waiting for. You can play a lifetime without to get anything or play for 5 minutes only and get it twice in a row. Nobody can tell.

@"Dreamy Lu.3865" said:A probability is, what its meaning tells without ambiguity: A probability = Nothing certain. Whatever the result of the calculation is, it does not tell when you will get what you are waiting for. You can play a lifetime without to get anything or play for 5 minutes only and get it twice in a row. Nobody can tell.

I think OP is well aware of that. Thing is, with the arguments in the recent posts here, it sounds like you shouldn't get suspicious if, after 300 dice rolls, you didn't get a single "6". Since the expected event should occur every 6th roll, it is reasonable to check if the die even has a "6" after 300 rolls without that event to occur. Nobody says that you must get a "6" with 6 rolls. The chance to get a "6" is always 1/6, and it's quite probable that you don't get one even after 20 or 30 rolls. It is highly unlikely though that you don't get a single "6" after 300 rolls while you expect that to happen 50 times. So unlikely in fact that you can assume that there is no "6" on that die. There is a variance for the event of getting a "6", that's why we don't expect to have exactly 50 times "6" after 300 rolls.

And that is similar to what happens here. With the difference that I don't question if that Guild Ballista drops at all, but if the drop rates are equal. Although it is "possible" that OP is just unlucky while the drop rates are equal, it is highly "improbable". It becomes much more probable if we assume a much lower drop rate for Guild Items. It's more reasonable to dismiss the premise of equal drop rates than to assume OP has really bad luck.

[edit] Alright, after the coffee took effect and woke me up, I looked into this again.

If you calculate it as: (1/21)^177 what you are doing is (1/21)(1/21)(1/21)...*(1/21) 177 times. It's not wrong, it's just not what you want to know.

Let's do this with a coin since it's easier to see. The event of "head" has an expected value of 0.5 or 50%. If you want to know how probable it is that you get 5 times Head after 5 coin flips, you multiply. 0.5 0.5 0.5 0.5 0.5 = 0.03125 or 3.125%. You calculated how likely it is that you get "Head" five times in a row. And the result is roughly 3%. The chance to not get five "Head" in a row is 97%. That includes the cases where you got four times "Head", three times "Head", two times "Head" one "Head" and no "head".

What OP and others calculated was the chance to get 177 Guild Ballistas out of 177 chests. Of course that's highly unlikely. The chance to get at least 1 Guild Ballista cannot be calculated that way. Maybe I'm going to do that later on, for now I'll say that with 1/21 (if the drop rates are equal) you would expect 8.4 Guild Ballistas dropping from 177 chests. It's not that unlikely to not get a single one. The drop rate research in the wiki suggests that it's only 2% for Guild Ballistas, so the expected number of drops from 177 chests is only 3.5 Ballistas.

I'm a bit embarrassed that I didn't see that before, but it's not something I do on a daily basis.

According to my calculations, if you open 177 chests after there's only the ballista missing, the probability of getting it is about 99.982%.If you consider this a negative binomial case, the expected number of chests you need to open (again, after there's only the ballista missing) should be 21.(edited again for more silly mistakes)

@"Faaris.8013" said:If you want to know how probable it is that you get 5 times Head after 5 coin flips, you multiply. 0.5 0.5 0.5 0.5 0.5 = 0.03125 or 3.125%. You calculated how likely it is that you get "Head" five times in a row. And the result is roughly 3%. The chance to not get five "Head" in a row is 97%. That includes the cases where you got four times "Head", three times "Head", two times "Head" one "Head" and no "head".

While you are right, the probability of getting at least one ballista is still [1 minus Pr(getting no ballista)], which actually does amount to 99.982%.

@Airdive.2613 said:

@"Faaris.8013" said:If you want to know how probable it is that you get 5 times Head after 5 coin flips, you multiply. 0.5

0.5

0.5

0.5

0.5 = 0.03125 or 3.125%. You calculated how likely it is that you get "Head" five times in a row. And the result is roughly 3%. The chance to not get five "Head" in a row is 97%. That includes the cases where you got four times "Head", three times "Head", two times "Head" one "Head" and no "head".

No, that is the probability of not getting 177 ballistas in a row.

@Faaris.8013 said:

No, that is the probability of not getting 177 ballistas in a row.

That is [100% minus the probability of not getting a ballista].

If you describe it in terms of binomial, P(0) = C(0 out of 177)x(20/21)^177x(1/21)^0,where C is the number of ways to draw 0 ballistas out of 177 chests (equal to 1),(1/21)^0 is the probability of drawing a ballista 0 times (still assuming equal odds for every drop),(20/21)^177 is the probability of getting anything that's not a ballista 177 times.

Thus, for P(more than 0 ballistas) you have roughly 100% - 0.018% = 99.982%.

(edited for clarity)

@Faaris.8013 said:

@Airdive.2613 said:

@Faaris.8013 said:If you want to know how probable it is that you get 5 times Head after 5 coin flips, you multiply. 0.5

0.5

0.5

0.5

0.5 = 0.03125 or 3.125%. You calculated how likely it is that you get "Head" five times in a row. And the result is roughly 3%. The chance to not get five "Head" in a row is 97%. That includes the cases where you got four times "Head", three times "Head", two times "Head" one "Head" and no "head".

Getting "Not a ballista" 177 times in a row is the only way to not have a single ballista after 177 tries, so a perfectly relevant thing to calculate.

@Faaris.8013 said:

@"Dreamy Lu.3865" said:A probability is, what its meaning tells without ambiguity: A probability = Nothing certain. Whatever the result of the calculation is, it does not tell when you will get what you are waiting for. You can play a lifetime without to get anything or play for 5 minutes only and get it twice in a row. Nobody can tell.

Actually, a better illustration would be rolling a 21-sider 177 times and not getting a 21. (Or, using dice that actually exist, rolling a 20-sider 160 times and not getting a 20. The odds are against that, but it's a lot closer than rolling a 6-sider 300 times.

@Airdive.2613 said:According to my calculations, if you open 177 chests after there's only the ballista missing, the probability of getting it is about 99.982%.If you consider this a negative binomial case, the expected number of chests you need to open (again, after there's only the ballista missing) should be 21.(edited again for more silly mistakes)

You said that SO much more easily than my first post. Thank you!

(Mathematicians may know math and even probability and statistics, but they don't necessarily know how to articulate that knowledge. :))

As mentioned above, you'd be making this post if any of the 21 were missing after 177 opens; with equal odds for each, you'd expect that to happen 0.373% of the time. Improbable, but also something I would expect to happen to someone in the game.

I hope you saved all the results though, because prior evidence implies they are not an even mix. Taking the point estimate from the data on the wiki assuming equal probability for guild siege, the probability of missing at least one of the guild siege after 177 open is... 8.32%

So they probably don't have equal drop rates, and you got somewhat (but not exceptionally) unlucky.

The odds are 99.983452215% in favor of the odds of getting the Guild Ballista not being 4.8%.

@"Ensign.2189" said:As mentioned above, you'd be making this post if any of the 21 were missing after 177 opens; with equal odds for each, you'd expect that to happen 0.373% of the time.

I'm not sure this is correct.My thinking would be that regardless of which one unique item is missing, if you choose the starting point in our calculations as "there is only one unique item missing", then the event "there is no [item] drop in 177 trials" is exactly the same. Point being, as you're opening your chests one by one, there will always be "one last item" missing.Intuitively, it just shouldn't take you this many chests to get your desired last item. (I'm not judging the design itself, just talking about numbers.)

@Gudy.3607 said:The maths are correct, assuming an even probability distribution for all 21 drops. Is it ridiculously unlucky, though? Maybeeee. If we assume an active player base of just 100,000 people, there are 16 others just as unlucky as you. With a million players, there would be 165 of you poor sods.

LOL.. I never remembered my Math teacher teaching the art of probability like this. Maths would of been so much more fun for all :)

As for your works OP.. I think you have the workings correct but the ultimate factor in it all is that only ANET know if the probability of each of the items dropping is equal... without that it comes down to lap of the RNG gods.I suggest buying a pair of Y fronts, place them on your head equi spaced between your eyes and run around your PC chanting I'm not worthy for an hour.. works every time for me cos I always seem to get the whatsamacallit for the thingamajig that I seem to remember wanting years ago when I first thought it was the beesall and endsall of my existence... my head hurts now :(

@"Airdive.2613" said:I'm not sure this is correct.My thinking would be that regardless of which one unique item is missing, if you choose the starting point in our calculations as "there is only one unique item missing", then the event "there is no [item] drop in 177 trials" is exactly the same. Point being, as you're opening your chests one by one, there will always be "one last item" missing.Intuitively, it just shouldn't take you this many chests to get your desired last item. (I'm not judging the design itself, just talking about numbers.)

The probability of not getting a guild ballista after 177 trials is 0.0178%, as mentioned above. However the probability of not getting a guild treb is also 0.0178%, and probability of not getting a guild catapult, etc. So if you were looking for the probability of not having a complete set after 177 opens, that would be 1-(1-(20/21)^177)^21, which is the .37% number above.

@evilsofa.7296 said:The odds are 99.983452215% in favor of the odds of getting the Guild Ballista not being 4.8%.

Well, if the alternative is the point estimate from the wiki data (2.29% chance of getting a GB), the Bayes factor between that and the null (equal chance) is 93.73; so about a 98.944% probability of the null being wrong. :)

@Ensign.2189 said:

@"Airdive.2613" said:I'm not sure this is correct.My thinking would be that regardless of which one unique item is missing, if you choose the starting point in our calculations as "

there is only one unique item missing

", then the event "

there is no [item] drop in 177 trials

" is exactly the same. Point being, as you're opening your chests one by one, there will always be "one last item" missing.Intuitively, it just shouldn't take you this many chests to get your desired last item. (I'm not judging the design itself, just talking about numbers.)

Let's assume the (this) thread is created in the event of "the last one item not dropping from 177 chests", which it is basically about. Then in terms of the probability of the existense of this thread there is no difference which one item that is, because we initially assume the OP was satisfied with the drop rate of the previous 20 items. It's only the word "ballista" that's going to change.I believe that if you want to calculate under your assumptions, you should take into consideration the whole experiment from the beginning, which, if I understand OP rightly, might have included more than just 177 chests. Because this thread might as well not have come into existence if not for the rare occurrence with "the last item desired" in particular.