The Gold to Shard converter is very generous

General Discussion
1

TL:DR You get an average of 83 shards per 3k gold spent BEST CASE SCENARIO vs 100 shards per 2k gold spent using the gold to shards converter.

The Gold to Shards converter is the fastest and most efficient way to get shards.

This is going to be a very mathematical post. If you’re allergic to math, you’re really not going to be able to refute what I’m about to say. If I did the math wrong, please, by all means, point it out. I’m not perfect.

Without further ado, I’m going to prove why the current proposed conversion rate of 500 shards for 10,000 gold is very generous.

First, we need to establish a few things.

  1. This assumes the best case scenario, that you own ALL items in the game.
  2. This also assumes the following probabilities are correct. (Which they’re probably not, but they’re close)
  3. This ignores the 18 chest Legendary pity timer as this would unnecessarily complicate the math.
  4. I am ignoring shard drops. This too will unnecessarily complicate the math because instead 35 permutations I will now have 126 permutations. Besides, the droprates of shard drops is unknown anyway.

The droprates for items in Loot Chests is as follows.

  • Legendary - 1/17 Chests
  • Epic - 1/4.5 Chests
  • Rare - 1/1 Chests

This makes the droprates as follows (On a per item basis, there are 4 items per chest).

  • Legendary 1/17/4 = 1/68 = 1.47%
  • Epic 1/4.5/4 = 1/18 = 5.56%
  • Rare 1/1/4 = 1/4 = 25%
  • Common 1 - (1/68 + 1/18 + 1/4) = 104/153 = 67.97%

Finally, the shard conversion rates for each item are as follows.

  • Legendary = 400 Shards
  • Epic = 100 Shards
  • Rare = 20 Shards
  • Common = 5 Shards

The weighted average can now be found easily.

4(400(1/68)+100(1/18)+20(1/4)+5(104/153)) = 79.35

This means there is an average of 79.35 shards per roll. When you strategically reroll, you can raise this average as you usually accept a value of 50 or more.

Now which would you rather do? (Remember, this assumes you own ALL items in the game)

Spend 10,000g on 3 chests with 1,000g to spend on rerolls or get a guaranteed 500 shards for 10,000g?

Now the rest is just for fun.

With 4 possible item types, and 4 rolls per chest, there are 35 permutations which can be derived from the following formula.

(r+n-1)!/(r!(n-1)!)

However, with rare chests, one permutation can be ignored, which is the case of 4 common items (Since you are guaranteed at least one rare) which leaves us with the following 34 permutations. (Compliments of mathisfun.com because there’s no way I’m walking through the list of unique combinations manually)

{L,L,L,L} {L,L,L,E} {L,L,L,R} {L,L,L,C} {L,L,E,E} {L,L,E,R} {L,L,E,C} {L,L,R,R} {L,L,R,C} {L,L,C,C} {L,E,E,E} {L,E,E,R} {L,E,E,C} {L,E,R,R} {L,E,R,C} {L,E,C,C} {L,R,R,R} {L,R,R,C} {L,R,C,C} {L,C,C,C} {E,E,E,E} {E,E,E,R} {E,E,E,C} {E,E,R,R} {E,E,R,C} {E,E,C,C} {E,R,R,R} {E,R,R,C} {E,R,C,C} {E,C,C,C} {R,R,R,R} {R,R,R,C} {R,R,C,C} {R,C,C,C}

To calculate the probability of a combination, you simply take and multiply the probability of each item in the combination.

For example, getting 4 Legendaries is a (1/68)^4 probability which is 0.0000005%. If you can’t guess, that’s very very low.

If you’re interested, here are the probabilities an shard amounts for each combination.

Combination Probability Shards
LLLL 0.00% 1600
LLLE 0.00% 1300
LLLR 0.00% 1220
LLLC 0.00% 1205
LLEE 0.00% 1000
LLER 0.00% 920
LLEC 0.01% 905
LLRR 0.01% 840
LLRC 0.04% 825
LLCC 0.06% 810
LEEE 0.00% 700
LEER 0.01% 620
LEEC 0.04% 605
LERR 0.06% 540
LERC 0.33% 525
LECC 0.45% 510
LRRR 0.09% 460
LRRC 0.75% 445
LRCC 2.04% 430
LCCC 1.85% 415
EEEE 0.00% 400
EEER 0.02% 320
EEEC 0.05% 305
EERR 0.12% 240
EERC 0.63% 225
EECC 0.86% 210
ERRR 0.35% 160
ERRC 2.83% 145
ERCC 7.70% 130
ECCC 6.98% 115
RRRR 0.39% 80
RRRC 4.25% 65
RRCC 17.33% 50
RCCC 31.41% 35
CCCC 21.35% 20

Anyway, there you go. Remember in the table that 0.00% is not actually 0%, it’s just too low to display with 2 decimals.

EDIT: I just realized that I forgot to account for multiplicity. For example, there are 24 different ways you can get a LERC chest. I will have to update the probabilities. Give me a few minutes to do this.

EDIT2: Fixed the probabilities, they now properly account for multiplicity.

EDIT3: Weighted calculation is wrong. The average can’t be 19.8 when the least amount of shards is 20. I must have done something wrong.

EDIT4: I forgot to multiply by 4 for 4 items per chest, lol.

3 Likes
2

Too much!

┻━┻ ︵ヽ(`Д´)ﾉ︵ ┻━┻

6 Likes
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No flipping tables, that is against the rules.

1 Like
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┬──┬ ノ( ゜-゜ノ) :sweat_smile:

8 Likes
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Thank you, sir.

Order has been restored to the universe.

1 Like
6

Is it Gems or Shards? Or is there an equivalence I don’t know about?

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Sorry, typo, that should be shards.

I was in a hurry, there were something like 200 formulas I had to write to do those calculations. I didn’t proof read the thread well enough before I posted it.

2 Likes
8

Fair enough.

One other question, ignoring how you got a common chest in the first place (brawl say), assuming you have 3 reroll (1500 gold) and that you would reroll unless you got over 500 shards, what are your chances of getting 500+ from 1 chest?

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So you’re asking what are your chances of getting a chest with a value of 500+ in 3 rerolls?

That’s an interesting math problem, let me think for a minute how to compute that.

Alright, I figured it out. You need to split this problem into two parts.

First, what is the probability of getting a chest with a value of 500. That part’s easy, simply add up all of the probabilities with shard values above 500. (I’m using excel)

That’s 1.03% of the time, you’ll have a roll that’s valued above 500.

Now since this is a Bernoulli trial, you use a binomial distribution.

nCr(p)^n(1-p)^(n-r)

In this case in particular, since we want at least one chest, this means we’re calculating 1-Chance of Zero chests valued at 500+.

1-binompdf(3,0.0103,0) = 1-.9694 = 3.05% chance to get a chest valued above 500 shards.

EDIT: Discourse does weird things sometimes…

1 Like
10

Yeah.

The other thing I that jumps out at me is the 18 Legendary pity timer. I didn’t know that was a thing.

Assuming that’s 1 in every 18 viewings, that’s every 6th loot chest assuming you reroll each chest 3 times.

Again ignoring how you got the chests, that’s 1500 gold x 6 or 9000 gold for at worst 20 + 20 + 20 + 20 + 20 + 415 = 515 shards. But will often be better than that.

It seems to me that if you have the time or Blizz has an event like the toy event, farming loot chests and rerolls will give you a better return on your gold.

11

I’m not 100% sure how the pity timer works, but it increases the chances of a legendary drop after the 18th chest (Not reroll AFAIK) but doesn’t guarantee one on the 18th. It just rapidly increases the chance until one drops, then it resets.

12

Ah okay. So the rest of that is wrong.

Going by your 3.05% for 500 shards formula, I calculate that after 6 loot chests there’s a 19.24% chance that 1 of them would be over 500 shards. Still not great.

Still I might run your second formula over some arbitrary points on your table and see what an expected return is on 6 loot chests with 3 rerolls = 9000 gold = 500 shards.

I know that’s not exact, but it’s close enough given you can’t spend 9000 to get 450 shards.

1 Like
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Keep in mind, this assumes you own EVERYTHING in the game so EVERY item converts to a shard.

1 Like
14

Yah, but that’s for assigning some sort of value to the chest results so we can actually compare 500 shards to rolling a loot chest. Isn’t it?

Obviously a legendary skin you don’t own is worth 1600 shards to you, not 400.

15

I suppose you can think of it that way, but you’re going to complicate the math real quick if you start going that deep. You’d have to take inventory of EVERYTHING you own, then multiply that ratio by the drop chances and go from there. It won’t match the probabilities I provided in the OP.

1 Like
16

the chests you buy with gold are Rare chests, you are unable to get 4 commons in the chest.

:stuck_out_tongue:

17

I thought I noted in the OP that you throw away the result of CCCC and now have 34 permutations just below that?

If I didn’t, I meant to.

Yeah, I did mention it.

Just take the probability of CCCC and add it to CCCR since it will force one common to be a rare.

I don’t think it starts with one rare and then rolls 3 times because that would be a worse scenario for the player. I think it just overrides one of the commons with a rare if there are no rares.

If it forces a rare, then take the probability of the result without the rare and add it to the one with the rare.

CCCC → RCCC
ECCC → ERCC
LCCC → LRCC

et cetera, so I guess there are even less permutations. But it really depends on how it works under the hood.

18

I probably just missed it.

It does not, it just forces that one item cannot be common. So LCCC is valid in a rare chest.

1 Like
19

Bro are you saying theres a shard converter in Ptr right now? I missed that…

20

I wasn’t 100% sure, so basically I was right, there are 34 instead of 35 permutations and CCCC->RCCC since every other permutation contains at least one item of rare or higher. So now the probability of an RCCC chest is 21.35% + 31.41% = 52.76% instead of 31.41%.

This barely shifts the average up to 82.54 instead of 79.34.

Rare chests provide barely any difference in shard income.