Optimising your chances on rolling rarity (new table explained)

[Petra Arbeiter]

Motivation

I keep seeing lots of incorrect statements about the chances of rolling up your weapon on the new upgrade table. Given a success rate of 5%, quite a few people have said that ten rolls will give you 100% chance. Of course that's not at all how probability works. But the actual numbers may be surprising.

We know that when we roll a die we have a 1 in 6 chance of getting a 5. Rolling it six times in no way guarantees we get a 5. In fact, you could roll that die 100 times and not get the 5 you want. People are sinking all sorts of arkforge into weapons this way.

I want to help people save cash, so I thought I would calculate the odds correctly. And furthermore, I will figure out which of the payment options is the best. This will depend on how risk averse you are. Or how lucky you feel. But at least you will be armed with facts.

This post will explain the mathematics. I will present the resulting tables and conclusions in follow-up posts.

The simple stuff

A probability is given in percent but we need to express it as a decimal to perform any calculations. For any single event, the chance of success plus the chance of failure will always add to 1. So, if our success rate is .05, our failure rate is 1-.05 = .95. This is simple enough, but key to everything that follows.

What we want to know is the cumulative chance, after a certain number of rolls, of getting to a total chance. For example, say you are happy having a 50-50 chance of upgrading your gun. This is the same as a cumulative total of .5. How many times would you need to spend 100 arkforge at 5% success rate to reach this? How much will you spend in total?

Trion has given us all sorts of options. We can try our luck at 5%, or 10%, or 15%, etc., all the way up in 5% increments. Or we can slap down 3200 arkforge and have 100% chance. Which makes the most sense?

I'll give two explanations of how the odds are calculated.

Explanation 1

Consider what the cumulative chance of success is after the first few rounds.

After one round our probability of success is .05. If we succeed, we stop rolling, because there is no reason to continue. We won! Thus, the chance of success after one round is .05 or 5%.

Consider the more likely outcome. Our chance of failing round one is .95. In this case, we try again. On round two, our chance of success is .05, so the cumulative chance to get here is .95 * .05. This is the important bit. We had to have first failed and then succeeded. We need to multiply together the two probabilities.

The result is 0.0475, which is our chance of success in the second round. Add this to the round one total, to get a running total. Chance of success after two rounds is .05 + 0.0475 = 0.0975 = 9.75%. Our chance has almost, but not quite doubled.

For round three, we must have failed twice before (in rounds one and two). Our cumulative chance of success is .95 * .95 * .05 = 0.045125. The incremental chance is smaller than before. We add this to the previous result, to get the running total. Chance of success after three rounds is 0.0975 + 0.045125 = 0.142625 = 14.26%.

The pattern should now be obvious, and we can write it up in code.

Explanation 2

Consider what the cumulative chance of success is at a given round, say round five. The chance of success is 1 minus the chance for failure (as always). So, let's calculate the chance for failure, since it works out to be simpler.

We fail on round five if all five rolls fail. So that's a probability of .95 for the first roll, .95 for the second roll, same for the third, fourth, and fifth. We multiply all these together to get the result. That's .95 times itself 5 times, otherwise known as .95 to the power 5.

The chance to succeed is therefore given by: 1 - (.95^5)

It's a wrap

Both these methods will return the same results. They are simply two ways of looking at the binomial distribution.

[Petra Arbeiter]

I wrote a short Python program to do the grunt work. I decided to try the 5%, 10%, 15%, 20%, 25%, and 30% odds, but then skip in increments of 10%. I doubt the values in between make much difference. And I figured you don't want to be reading tables all day.

Trion has chosen a confusing way of presenting the costs. I am assuming that the base cost of 100 gets added to the incremental cost each time. So each attempt at 10% costs 175 arkforge, each attempt at 15% costs 250 arkforge, etc. If this is not true, please tell me! I can correct the numbers and run off the tables again. No, I am not going to actually try it and see.

I terminated each run when the chance for success hit 100% or when the cost hit 3200. I think it's obvious why. If it's going to cost more than what you can pay to get an upgrade with certainty, then you are certainly doing something wrong!

OK here's the first table, for 0.05 success @ 100 arkforge a throw. The first column is how much you spend. The second column gives you the cumulative chance of success.

  100   5.00
  200   9.75
  300  14.26
  400  18.55
  500  22.62
  600  26.49
  700  30.17
  800  33.66
  900  36.98
 1000  40.13
 1100  43.12
 1200  45.96
 1300  48.67
 1400  51.23
 1500  53.67
 1600  55.99
 1700  58.19
 1800  60.28
 1900  62.26
 2000  64.15
 2100  65.94
 2200  67.65
 2300  69.26
 2400  70.80
 2500  72.26
 2600  73.65
 2700  74.97
 2800  76.22
 2900  77.41
 3000  78.54
 3100  79.61
 3200  80.63

What can we learn from this?

First, we get less for our money the more rounds we go. Yes, we are still increasing our chance of success, but by less and less. This is what the house doesn't want us to know. The house always counts on the gambler not being too smart.

To have a 50-50 chance of success, we need to spend 1400.

After spending 3200 arkforge, you still have a one in five chance of failure. Obviously that compares badly with simply paying the 3200 in the first place.

So, how risk averse are you? Will you take a chance on the 5% table?

[Petra Arbeiter]

Here are the remaining tables. The first entry makes it clear what odds are being used. "175 10.00" means it costs 175 arkforge to get a 10 percent chance of success.

175  10.00
  350  19.00
  525  27.10
  700  34.39
  875  40.95
 1050  46.86
 1225  52.17
 1400  56.95
 1575  61.26
 1750  65.13
 1925  68.62
 2100  71.76
 2275  74.58
 2450  77.12
 2625  79.41
 2800  81.47
 2975  83.32
 3150  84.99

250  15.00
  500  27.75
  750  38.59
 1000  47.80
 1250  55.63
 1500  62.29
 1750  67.94
 2000  72.75
 2250  76.84
 2500  80.31
 2750  83.27
 3000  85.78

350  20.00
  700  36.00
 1050  48.80
 1400  59.04
 1750  67.23
 2100  73.79
 2450  79.03
 2800  83.22
 3150  86.58

450  25.00
  900  43.75
 1350  57.81
 1800  68.36
 2250  76.27
 2700  82.20
 3150  86.65

575  30.00
 1150  51.00
 1725  65.70
 2300  75.99
 2875  83.19

850  40.00
 1700  64.00
 2550  78.40

1175  50.00
 2350  75.00

1550  60.00
 3100  84.00

1965  70.00

2400  80.00

2845  90.00

[Petra Arbeiter]

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.

[Ashlocke]

Awesome write up, thank you for this.

[YahootsMagoondi]

[Fuggleer]

OP, your first name wouldn't happen to be Greg would it?

I absolutely love and applaud your efforts here. Unfortunately, as I am sure you know, this might as well be written in Martian to the average gamer.

[Chump Norris]

My head hurts, time for a few shots.

[richardkrainium]

remember this is trion math so if ya forgot to subtract the 2 and carry the banana, all that work was for nothing

[Smokey Black]

Wish I would of dropped 1500 all at once instead of trying to be cheap.