Optimising your chances on rolling rarity (new table explained)

[dirt33gravy16]

OK, how to analyse this? I think the simplest way is to set a probability threshold you are comfortable with. Say you want a .6 chance of success overall. Is the .1 probability table better than .4 table? Depends on which has the lowest cost.

Here is the summary, in my best ASCII art. The row headers indicate the probability you want, from .1 (10%) to 1.0 (100%). The column headers indicate the table that's being used.

      .05 |  .10 |  .15 |  .20 |  .25 |  .30 |  .40 |  .50 |  .60 |  .70 |  .80 |  .90 |  1.0 | 
     ----   ----   ----   ----   ----   ----   ----   ----   ----   ----   ----   ----   ----
.1 |  200 |  175 |  250 |    - |    - |    - |    - |    - |    - |    - |    - |    - |    - |
.2 |  500 |  525 |  500 |  350 |  450 |    - |    - |    - |    - |    - |    - |    - |    - |
.3 |  700 |  700 |  750 |  700 |    - |  575 |    - |    - |    - |    - |    - |    - |    - |
.4 | 1000 |  875 | 1000 | 1050 |  900 |    - |  850 |    - |    - |    - |    - |    - |    - |
.5 | 1400 | 1225 | 1250 | 1400 | 1350 | 1150 |    - | 1175 |    - |    - |    - |    - |    - |
.6 | 1800 | 1575 | 1500 | 1750 | 1800 | 1725 | 1700 |    - | 1550 |    - |    - |    - |    - |
.7 | 2400 | 2100 | 2000 | 2100 | 2250 | 2300 | 2550 | 2350 |    - | 1965 |    - |    - |    - |
.8 | 3200 | 2800 | 2500 | 2800 | 2700 | 2875 |    - |    - | 3100 |    - | 2400 |    - |    - |
.9 |    - |    - |    - |    - |    - |    - |    - |    - |    - |    - |    - | 2845 |    - |
1  |    - |    - |    - |    - |    - |    - |    - |    - |    - |    - |    - |    - | 3200 |

What does this tell us? Say you are happy with 30% total chance of success. Go down to the third row (.3) and read off the numbers. Which is the cheapest? 575 arkforge. That's in the .3 table. So, you should just go ahead and roll once at 30%.

In fact, it turns out that, in most cases, you should just buy once on the table that corresponds to the percentage chance you want. There are two minor exceptions.

If you want a 50% chance, then roll twice on the 30% table, to save 25 arkforge on average (1150 versus 1175). Besides, this actually gives you a 51% chance.

If you want a 60% chance, then roll six times on the 15% table, saving 50 arkforge on average (1500 versus 1550). This actually gives you a 62.29% chance.

Of course, all of this requires discipline. In reality, how many people are going to roll 50% once and then accept the result? Most would roll again, increasing their chance to 75% and their outlay to 2350. But that is counter-productive. It would have been better to roll once on the 80% table to begin with (for 2400 arkforge).

It's all psychology. The only reason the lower percentage options exist at all is to sucker you into using them. They don't make any mathematical sense.

Avoid temptation. Save up for the percentage chance you want. Then stick with it.

I hope that's clear. Ask any questions and I'll answer them next time the servers are down.

[dirt33gravy16]

thank this was very informative but ill prob. shot for the 100% chance

[Big Jack Shepard]

I read most of the first page and skimmed the rest of the thread. So, sorry if you already stated this but, which % are you using Petra?

[Cobra Crusher]

He is not talking about averages, he is calculating odds. 2 totally different things.

It's like sayings that if I throw free throws and get 0 for 10 and a pro throws 10 for 10. The average for us both is 50/50 but the odds of me making 1 is not 50/50

Throw free throws? lol Not into sports are you?

[Ray8888]

That's not just bad sport terminology but a bad metaphor too.

[ArthriticNed]

Each try is its own action and has absolutely zero effect on the next try or any other. Each 5% try gives a 5% chance to get your upgrade. Doing multiple in a row at 5% doesn't gradually increase the next try, its still 5%.

[Mrdr]

Each try is its own action and has absolutely zero effect on the next try or any other. Each 5% try gives a 5% chance to get your upgrade. Doing multiple in a row at 5% doesn't gradually increase the next try, its still 5%.

You're correct, but he's calculated the odds of multiple attempts at 5%, or if you prefer, the probability of landing a supreme at varying chances over time.

[tpollett]

To the op: excellent analysis though you forgot to include the additional 100 arkforge base cost for 100%. Not trying to nitpick your work as I think your table will remain accurate. I'm not positive on the other numbers but I think they are all correct; I know up to 50% they are. This table shows that it is nearly always best to use the 15% option for 250 arkforge; which is what my gut was telling me, but I usually ignore my gut as it's an idiot.

ArthriticNed: you are correct that each individual chance is 5%. This is often used in the form of "flip a coin ten times, the first nine come up tails, what is the odds that the tenth flip will be heads?" The answer is 50/50. But that is not what the op is doing here. He is asking, what are the chances that flipping a coin ten times will result in tails in all ten flips. That's a different question entirely.

[GahdolfMiLahwn]

ArthriticNed: you are correct that each individual chance is 5%. This is often used in the form of "flip a coin ten times, the first nine come up tails, what is the odds that the tenth flip will be heads?" The answer is 50/50. But that is not what the op is doing here. He is asking, what are the chances that flipping a coin ten times will result in tails in all ten flips. That's a different question entirely.

Actually, the question is if you flip a coin up to ten times, and stop when you get a head, what is the chance you ended up with a head.

And the results at the 5% level of chance are highly sensitive to how close to 5% that chance actually is. If the advertised 5% is actually 8% then the crossover is a lot sooner, if its actually 3% then it's even more of a ripoff.

And I bet Trion did not do the testing to 95% confidence

[whit]

I upgraded 100000 guns a number of times last night. I am very rusty at stats but still remember enough C to write a simple simulator. My main reason to do this was to answer my own question, if 10000 guns are upgraded at 5% chance how many goes on average does it take for each upgrade. Having run the simulator a number of times at 5% for 10000 and 100000 guns, the answer (unsuprisingly) was ~20. At 10% it was ~10 goes etc.

Anyway, I would like to forget that I even asked that question and concentrate on more important matters. The numbers I got from the simulator back up Petra's analysis. I did it for all the percentage chances and all were very close to Petra's probabilities. Below is an example for 100000 guns at 5% chance. Compare the cumul(ative) and Petra columns.

go    AF  count  cumul  petra        go    AF  count cumul   petra
 1   100  5065   5065    5.00        17  1700  2165  58263   58.19
 2   200  4699   9764    9.75        18  1800  2015  60278   60.28
 3   300  4582  14346   14.26        19  1900  1980  62258   62.26
 4   400  4255  18601   18.55        20  2000  1911  64169   64.15
 5   500  4066  22667   22.62        21  2100  1843  66012   65.94
 6   600  3748  26415   26.49        22  2200  1716  67728   67.65
 7   700  3714  30129   30.17        23  2300  1637  69365   69.26
 8   800  3441  33570   33.66        24  2400  1465  70830   70.80
 9   900  3403  36973   36.98        25  2500  1443  72273   72.26
10  1000  3216  40189   40.13        26  2600  1402  73675   73.65
11  1100  3076  43265   43.12        27  2700  1411  75086   74.97
12  1200  2852  46117   45.96        28  2800  1228  76314   76.22
13  1300  2679  48796   48.67        29  2900  1160  77474   77.41
14  1400  2584  51380   51.23        30  3000  1070  78544   78.54
15  1500  2436  53816   53.67        31  3100  1099  79643   79.61
16  1600  2282  56098   55.99        32  3200   995  80638   80.63

I think we all agree Petra's analysis is excellent, however, he has kind of put the onus back on us to choose the target percentage (not to be confused with upgrade chance) that we are happy with. So as a simple alternative, I would like to ask that if you had to choose one upgrade chance from the upgrade menu and stick with it, what would it be ? Like tpollett, I think 15% is the best bet.