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Game Mechanics: percentages on cards

Feb 11, 2009
I think something needs to be cleared up as I have seen several posts that bring up the percentage on the cards. This percentage is a percentage of success not the rate of success. They may sound the same but they are not. Many have tried to track this percentage over time and in doing so have gained evidence that the percentage is off. They are wrong. That is a rate of success. What the percentage on the cards shows you is the chance of success. This can't be tracked over time because it is a single instance percentage. It is not saying thatthe card will work 7 out of 10 times but that at that moment in time, that card has a 70% chance of working.

Two examples outside of this game will explain this further. Take a quarter. It has a 50% chance of coming up heads. If you flip it 100 times, it is entirely plausable that you would never ever see a head. That is because the 50% is a chance of success. It does not guarantee an eventual result only the odds of getting one.

The other example would be the old pencil and graph paper role playing games. They use a couple of dice that have the same function as the code behind the percentage. In those games if you have a 70% chance of success you role the dice and any result between 1 and 70 grants you success.

I hope this clears up the confusion.

Apr 15, 2009
Well I guess if you want to be technical, they are the same. The long run average should still be 70%. (theoretically) The part that people get confused on is if you have a 1 out of 7 chance at casting, you can fizzle 21 times in a row before it becomes *almost* statistically impossible to fizzle again. Sure, you could fizzle one more time for 22, but the odds of that happening, for all pratical purposes, is impossible.

But yes, long run averages are just that... long runs. If you only cast a spell 10 or 20 times, it may not even be close to representing that 70% number.

Similar to your coin scenario, sure nothing is stopping you from getting 100 heads in a row, but after the 7th toss, you are pretty much going to get a tails.

I completely agree though. Many games will have boss drop percentages for items. Some item will have say a 1 out of 10 chance to drop, and you always get posts and complaints of, "OMG I killed said dude 18 times! Where is my item?!?"

Statistics are fun.

Jan 20, 2009
kordach wrote:
Take a quarter. It has a 50% chance of coming up heads. If you flip it 100 times, it is entirely plausable that you would never ever see a head.

This is entirely false.

While you would expect to get 50 heads and 50 tails, certainly a small deviation from these numbers would be expected. Given a fair coin, however, it is not "entirely plausable" to get 100 tails, in fact that it's so unlikely that you would be able to say with >99% certainty that said coin is in fact not fair.

In fact your entire argument is faulty. Saying that you can't track a single instance percentage of success over time is just silly. If I have a coin that I expect to flip heads 85% of the time and I take 100 observations of this coin, if my observations are very much off from 85 heads and 15 tails, I would in fact suspect that this coin doesn't flip heads 85% but some other number. If I take enough of these observations I can say that with reasonable certainty.

You need to brush up on your statistics, my friend.

Feb 11, 2009
I agree that eventually the rate of success will catch up, or come close, during that time the rate of success is usually lower than the chance of success, a 70 % chance usually yeilds around a 60% success rate. I think it happens mainly to annoy math people. lol.

Typically to get close enough to call them equal or the same, the sample needs to be far greater than the 50 to 100 that people use to foster conspiracies. I would say a good sample, if anyone is really interested in proving that KI is trying to get you to buy crowns would be using the same card against the same enemy in the same area for a 1000 or so tries. The storm card rate will still probably not be 70% but it will keep them busy and off the boards for about a month.

May 02, 2009
The % that you see on the card is probably far from the final chance that your spell will fail or succeed.

I have noitced that certain combinations of spells will have a VERY high fizzle rate. Even to the point that I am surprised if the spell does not fizzle. Such an example is if I place a blade on myself, and then a damage shield on my enemy, I would bet that more time than not, if on the third round I cast a three pip spell on the enemy, the spell will fizzle.

It happens so much that I noticed it happening. So the 70-80-85 etc that you see on the card, what does this really mean? What other factors are behind the scenes that we don't see, or aren't aware of which increase or decrease the fizzle rate? Is it a fair 70% chance? In my experience it is not. I am supposed to have an 85% chance that my spells will not fizzle, yet it 'seems' like it happens a lot more than 15% of the time, maybe 25-30%.

Are the % chances the same for one monster as it is VS 2 or 3 monsters? Are they the same in the arena as they are out in the environment?

In all honesty though, I don't think the number on the card reflects the actual fizzle rate.

Apr 15, 2009
There is also the idea that people will remember certain fizzles much more strongly than the successful casts. For example, you can be casting centaur all day long, and it is working like usual, but you have no reason to take note of it. Then you get to a boss, cast several different blades and traps, you are all ready for that huge centaur hit that is going to finish the boss off, and FIZZLE! What bad luck, after all, it had a 90% chance to land. You then finish the boss off and continue on your way.

Then you get to another boss, and the same thing happens again. Now it seems like that 90% centaur is broken, after all you spent all that time casting blades and traps, and then it goes and fizzles, not once, but two boss fights in a row.

Such memories are much more prominent than most others. If you were to actually track ALL centaur casts, and not just notice the ones that effect you the most, then it probably falls in nicely with that 90% range.

I can say this is true for when I cast scarecrow. I don't usually use it much, but when I do, it is always suppose to be for that 'perfect' moment. You have shields up on multiple enemies, they are all low on health, you can finish them all off with just one cast. Every time that happens and fizzles, I always take note of it much more strongly than I would have if it just worked like normal.

Feb 11, 2009
Umm, no i dont need to brush up on my stats. It is not PROBABLE to get tails 100% of the time but it is PLAUSABLE. My entire argument is entirely sound. Over a vast timeline you may get close to the number on the card but the number on the card is the odds that particular card will succeed or fail. While the two stats are related, they are NOT the same.

YOur counter argument on the coin is faulty in that it is arguing a different argument. The coin has a 50% chance of coming up heads. I never said and its not true to suppose that a coin will come up heads 50% of the time. A coin has a 50% chance of coming up heads. You flip it and it comes up tails. You pick up the coin and its odds of coming up heads has not changed. the coin doesnt know it came up tails last time. It is a singular event. Because you got five tails in a row you should not expect to see a string of heads.

What I have explained is how the cards work. Like it or not, agree with it or not, this is how it happens and when you follow the stats over time as all these conspiracy zealots have, it only proves my point.

Dec 31, 2008
I think sirap's point about questioning whether the coin is fair is a good one. It has to do with how we decide if the cards really have the accuracy you see printed on them. There are several people currently claiming that the printed stats are not accurate. If you take a sample of the cards' behavior over some period of time, you can calculate how likely it is that you'd get such a sample if the odds really match the printed stats. If the likelihood of getting your sample from a correct set of cards is quite low, then you have a strong case for asking KI to fix the cards.

Unfortunately, most people's samples are a) not random (because they rely on memory, which is highly selective as Merak points out), and b) too small to make drawing them from a correct set of cards very unlikely.

Kordach, I'm sure you know this stuff. I'm just trying to move away from the coin example, where the underlying odds are already known, and onto the kind of problem where you use a limited sample to check whether the underlying odds match expectations or not.

Dec 02, 2008
There are several issues here that need to be cleared up. Some people in this thread are appealing to the Strong Law of Large Numbers, albeit without naming it explicitly. Loosely, if a random event is repeated enough times, the frequency with which something happens in the long run approaches the true probability of the event. More formally:

If X_1, X_2, ... are pairwise independent and identically distributed random variables with finite expected value m and S_n = X_1 + ... + X_n, then the sequence (S_1)/1, (S_2)/2, (S_3)/3, ... converges to m almost surely.

The Strong Law of Large Numbers guarantees that a sequence will eventually converge, but tells nothing about the rate of convergence. It doesn't rule out the possibility that the first trillion terms in the sequence could be unlikely to be anywhere near m, and it's not too hard to construct random variables that would make convergence very, very slow. The argument here is fundamentally about the rate of convergence.

Incidentally, there is also a Weak Law of Large Numbers, but the distinction (both between the assumptions and the results) is fairly subtle, and unless you've studied it, your intuitive notion is probably that of the strong law.

With stronger assumptions, we can get the Central Limit Theorem, which says not merely that the frequency converges to its true value, but how far away from the true value it tends to be. To the above theorem, add the assumption that X_1 has finite variance s^2 and we get that (S_n - nm)/s*sqrt(n) converges weakly to the standard normal distribution.

While not technically a result that says you must converge at least this fast, for a binary random variable with probabilities not near 0 and 1, the distribution will be very close to the standard normal distribution very quickly. More precisely, the probability that (S_n - nm)/s*sqrt(n) < k is very close to the integral with respect to x from negative infinity to k of the function e^(-x^2/2)/sqrt(2pi). If the X_1 takes on the value 1 with probability p and the value 0 with probability 1-p, then X_1 has variance s^2 = p - p^2. Thus, if you want to compute the probability that you'll be off from the true value by a given amount after a given number of trials, this gives you a very good approximation. The function e^(-x^2) has no elementary antiderivative, but one can compute very good approximations, such as via Taylor series.

Another issue is the distinction between something that cannot happen and something that is merely so unlikely that you'll realistically never see it. In this regard, usually the reason that people get the wrong answer is that they are asking the wrong question.

If you're just playing the game and happen to say, whoa, that fizzled four times in a row, what's the probability of that, then you could perhaps compute the probabilty that four particular spells would fizzle and find that it is very low. But that is asking the wrong question. If you cast a spell a thousand times, the probability that some four consecutive casts will fizzle may actually be quite high. The question that you should have asked is not, would these four particular casts all fizzle, but would there ever be some four casts that fizzle?

In some cases, the better question to ask is not, what are the chances that I'd personally see this, but what are the chances that some player at some point would see this? The probability of that can be much higher still: the probability that you will win the Powerball jackpot this year is very low, but the probability that someone in the country will win it is very high.

The only way around this is to do a controlled trial. Before you collect any data, you have to decide that you're going to do some particular action a fixed number of times, in controlled circumstances that make the probabilities not vary from one trial to the next. You have to decide what you're going to do (including the number of trials) before you start, and then record every single data point that comes. That prevents you from cherry-picking data points to see what you wanted to see, rather than what was actually there. Most humans cannot run a controlled trial, though, as they will lose interest and not record all of the data points, which makes the whole data set garbage. If you record every single fizzle, but fail to record a quarter of the times that the spell doesn't fizzle, then of course you'll end up concluding that the fizzle rate is too high.

If you compute the right probability and discover that the odds that you'll be as unlucky as you were are something like 1 in 3, then that's just a fluke. Either the true probability is what was claimed, or else your sample size was merely too small to tell the difference. If you do the proper computations and get a probability that you'd be at least as unlucky as you were and get a probability of 1 in a million, that's very unlikely to be a fluke. Most likely it's an error in either recording data or doing computations on your part, but it's still not a fluke.

Just because the probability of an event is very low doesn't mean that it can't happen. It can be improbable enough to safely assume that you'll never see it happen, though. Humans tend to intuitively overestimate the odds of very low probability events; this is why so many people play the lottery.

Still, in the technical sense, just because an event has probability zero doesn't mean it can't happen. For example, pick a real number uniformly at random on the interval (0, 1). Let's say you pick x. What is the probability that you would have picked x? For any e > 0, the probability that you'd have picked a number between x - e/3 and x + e/3 is 2e/3 < e. Since x is in this interval, the probability that you'd pick x is less than e for all e > 0. That is, the probability is zero.

To distinguish this sort of event from something that truly cannot happen at all, an event that occurs with probability zero is said to "almost surely" not happen. The name sounds really hand-wavey, but it does have a precise meaning.

May 19, 2009
OKAY...it's not that complex. I believe, while others agree with me, that this game cheats majorly (I love it, but it cheats). I've had several 100% cards (buffs, shields, minor attacks) fizzle although there were no spells to make my casts less likely to work. It has happened countless times, coincidentally when I'm in a dungeon, tower, or fulfilling a mission. Another coincidence is how my cards fizzle when I need them most. I might need one more hit with a card that I've been saving pips for and losing health because of it. All of a sudden *FIZZLE* and the enemy either winds up healing or killing me. Then I coincidentally can't find the card in my deck until I don't need it anymore. So I don't pay attention to the percentages. I just use the best strategy and hope that my 100% cards do what they are supposed to.
And the "completely plausible" coin flipping thing is insane. It may seem plausible since we are talking about probability, but that is most definitely not plausible. I could understand if the chances we closer to tails than heads. Say 90% vs. 10%, but 50/50 with all tails is not as plausible as you explained, sir. And to further justify my argument, you spelled plausible incorrectly.