Worst starting hands ever

[flyingcat]

Against Treldon in BL, playing Immorush...
1 nova, 3 spark, 3 immo... joy

[OldTrees]

Alright, working through the logic before spending time to collect stats/experiment please review and confirm the following: 

- Shuffle deck (30 cards for out test case)
- Draw 7 cards from 30-card deck for initial opening hand (not yet revealed to player in-game)
- Check for presence of at least one 0-cost card... Yes = reveal hand to player and start game... No - mulligan process (below)
* We only care to further investigate cases with no 0-cost cards in this initial opening hand
* (1) Tower and (29) unplayable card deck calculates to .76666 probability of needing a Mulligan process
* The mulligan process is 'invisible' to the player in-game
- If no 0-cost cards then draw next 7 cards from remaining 23-card deck for mulligan opening hand(1 Tower and 22 unplayable cards is 0.69565 probability to still have no 0-cost cards) (The mulligan hand is revealed to the player no matter what 0-cost cards are in the hand and the game starts)
** When the (1) Tower and (29) unplayable cards deck enters the mulligan process we need to search for instances of final opening hand containing no 0-cost cards and then review the FIRST SEVEN TURNS of this resulting draw for presence of that lone Tower.
** The M1s model is proven present in the Elements programming if the Tower is drawn on ANY of the first seven turns following a no 0-cost final opening hand.

Spoiler for Such as this one... I think this proves that EtG reshuffles the 23 cards after a Mulligan process/attempt.:

Elements Game Simulator – by Xenocidius
________________________________________
Enter Deck 1: http://dek.im/d/z54vkz555qz56ttz56tu710z47488pk
HP:   Marks:   Double draw: 
Enter Deck 2: http://dek.im/d/z54vkz555qz56ttz56tu710z47488pk
HP:   Marks:   Double draw:   Use AI deck instead   Level: 

Player 1 was dealt Mutation
Player 1 was dealt Momentum
Player 1 was dealt Micro Abomination
Player 1 was dealt Micro Abomination
Player 1 was dealt Mutation
Player 1 was dealt Micro Abomination
Player 1 was dealt Micro Abomination

Turn 2
Player 1 drew Bone Tower
Edit 1: Added spoiler with evidence I believe proves the reshuffle after mulligan process before starting game.

Yes, that is a perfect counterexample to model M1ns. This confirms that Xeno's simulator does not use M1ns.

[Faro]

I did a quick test agains AI1 with this

Spoiler for Hidden:

Hover over cards for details, click for permalink

Deck import code : [Select]

4sa 4sa 4vp 50a 52l 52r 56i 58v 59m 5c1 5c9 5fa 5io 5j2 5m6 5od 5oj 5rh 5s4 5up 5v0 61r 623 74g 7ee 7ee 7ee 7ee 7ee 7k2 8po

not random...

and I got this

Spoiler for Hidden:

Position in the deck of the first quantum pillar

16
19
1
1
23 L
1
1 B
1
1
15
1 B
1
19 L
1
1
1
1
16
1
1
1 B
1
23
1
17
15
19
19
1
1

*
1 = first hand
L = lost and draw N cards
B = both QP in first hand

I'll let you do the math, but at least 10 times out of 30 the mulligan happend  and i never draw a pillar in the first 14 cards (2 full hands). coincidence?

[OldTrees]

I'll let you do the math, but 10 times out of 30 the mulligan doesn't work and i never draw a pillar in the first 14 cards (2 full hands). coincidence?

That makes 52 so far. So what we need now is a calculation of correct sample size. How many cases of EtG games(not the simulator) that could have but failed to disprove M1ns are needed for the probability of M1s to be less than 0.5%?

Aka for a sample size of X games with no pillar in the starting hand, all X games having no pillar in the first 14 cards is less than a 0.005 probability under the M1s model? (Obviously if 1 of those X games showed differently it would disprove M1ns regardless of the others)

That way we could run X such games as a test of the EtG mulligan rules.

[rob77dp]

I did a quick test agains AI1 with this

Spoiler for Hidden:

Hover over cards for details, click for permalink

Deck import code : [Select]

4sa 4sa 4vp 50a 52l 52r 56i 58v 59m 5c1 5c9 5fa 5io 5j2 5m6 5od 5oj 5rh 5s4 5up 5v0 61r 623 74g 7ee 7ee 7ee 7ee 7ee 7k2 8po

not random...

and I got this

Spoiler for Hidden:

Position in the deck of the first quantum pillar

16
19
1
1
23 L
1
1 B
1
1
15
1 B
1
19 L
1
1
1
1
16
1
1
1 B
1
23
1
17
15
19
19
1
1

*
1 = first hand
L = lost and draw N cards
B = both QP in first hand

I'll let you do the math, but at least 10 times out of 30 the mulligan happend  and i never draw a pillar in the first 14 cards (2 full hands). coincidence?

Firstly, I'm not sure how you know when the mulligan was activated as it is "invisible".  One can determine when the mulligan was activated AND failed because you'll have an opening hand with no 0-cost cards.  Also, I count 11 instances of your first QP being beyond the 14th card.  The math regarding drawing BOTH QP's in the opening 7 cards is 0.0483 + (0.5861 * 0.0830) = 9.69%.  This lines up as 0.0969 * 30 = 2.91 expected instances compared to the 3 you experienced.  A-OK there!

The math also says that with a 30-card deck containing 2 zero-cost cards the probability of not drawing a QP in opening hand is the probability of no QP in the initial opening hand multiplied by the probability of drawing no QP on the mulligan hand (when not drawing a QP in the initial opening hand):
(1 - 0.4139) * (1 - 0.5257) = 0.2780, or 27.8% of the draws with such a deck will yield you an opening hand without a QP.  0.278 * 30 = 8.34 expected instances of no QP the opening hand in 30 game sample size.

Now, to go another 7 turns without drawing one of the (2) QP's in the deck is another iteration of the "mulligan process" - that of 2 successes in 23 size population with sample size of 7.
0.2780 * (1 - 0.5257) = 0.1319, or 13.2% of your games with such a 2 zero-cost 30-card deck will leave you QP-less 14 cards into the deck.  0.1319 * 30 = 3.957 expected instances of no QP in the first 14 cards in a 30 game sample size.  Your experience shows NO instances of drawing a QP outside your opening hand BUT within the next 7 cards - very strange indeed.  Also, the 11 games without QP in the first 14 cards is more than expected and the ELEVEN without a AP in the first 14 cards is much higher than the expected.  None of this shocks me as the sample size of 30 games is quite small for values this sensitive to sample size.

More experimentation is required to get us all to the bottom of this! 

[OldTrees]

Firstly, I'm not sure how you know when the mulligan was activated as it is "invisible".  One can determine when the mulligan was activated AND failed because you'll have an opening hand with no 0-cost cards.

Yep, a mulligan failing is the only sign a mulligan happened. Useful signal.

Now, to go another 7 turns without drawing one of the (2) QP's in the deck is another iteration of the "mulligan process" - that of 2 successes in 23 size population with sample size of 7.
0.2780 * (1 - 0.5257) = 0.1319, or 13.2% of your games with such a 2 zero-cost 30-card deck will leave you QP-less 14 cards into the deck.  0.1319 * 30 = 3.957 expected instances of no QP in the first 14 cards in a 30 game sample size.  Your experience shows NO instances of drawing a QP outside your opening hand BUT within the next 7 cards - very strange indeed.  Also, the 11 games without QP in the first 14 cards is more than expected and the ELEVEN without a AP in the first 14 cards is much higher than the expected.  None of this shocks me as the sample size of 30 games is quite small for values this sensitive to sample size.

More experimentation is required to get us all to the bottom of this! 

If I understand correctly:
0.5257 is the chance of drawing at least 1 pillar in the first 7 draws given a failed mulligan under the M1s model?
0.4743 is the chance of drawing no pillars in the first 7 draws given a failed mulligan under the M1s model.
0.4743 ^ X is the chance of drawing no pillars in the first 7 draws given a failed mulligan under the M1s model for X games in a row.
Does 0.005 = 0.4743 ^ Y define the number of games(Y) that would have a combined chance of less than 0.5% chance of occurring under the M1s model?
Y = Log_0.4743(0.005) = 7-8

That seems awfully low. I suspect I failed stats here.

[rob77dp]

Firstly, I'm not sure how you know when the mulligan was activated as it is "invisible".  One can determine when the mulligan was activated AND failed because you'll have an opening hand with no 0-cost cards.

Yep, a mulligan failing is the only sign a mulligan happened. Useful signal.

Now, to go another 7 turns without drawing one of the (2) QP's in the deck is another iteration of the "mulligan process" - that of 2 successes in 23 size population with sample size of 7.
0.2780 * (1 - 0.5257) = 0.1319, or 13.2% of your games with such a 2 zero-cost 30-card deck will leave you QP-less 14 cards into the deck.  0.1319 * 30 = 3.957 expected instances of no QP in the first 14 cards in a 30 game sample size.  Your experience shows NO instances of drawing a QP outside your opening hand BUT within the next 7 cards - very strange indeed.  Also, the 11 games without QP in the first 14 cards is more than expected and the ELEVEN without a AP in the first 14 cards is much higher than the expected.  None of this shocks me as the sample size of 30 games is quite small for values this sensitive to sample size.

More experimentation is required to get us all to the bottom of this! 

If I understand correctly:
0.5257 is the chance of drawing at least 1 pillar in the first 7 draws given a failed mulligan under the M1s model?
0.4743 is the chance of drawing no pillars in the first 7 draws given a failed mulligan under the M1s model.
0.4743 ^ X is the chance of drawing no pillars in the first 7 draws given a failed mulligan under the M1s model for X games in a row.
Does 0.005 = 0.4743 ^ Y define the number of games(Y) that would have a combined chance of less than 0.5% chance of occurring under the M1s model?
Y = Log_0.4743(0.005) = 7-8

That seems awfully low. I suspect I failed stats here.

My math-ing indicates Y = 7.103 in the above which would mean that after 8 games the likelihood of the M1s model drawing NO pillars in first 7 draws after a mulligan is less than 0.5%.

I just did my own test run with a 2-Tower 28 non-playable cards 30-card deck.  Results:

*15
7 (none in next 7 draws)
7 (none in next 7 draws)
2&4
3 (next on 4th draw)
4 (none in next 7 draws)
*22
1 (none in next 7 draws)
5 (next on 6th draw)
*25
*22
2 (none in next 7 draws)
1 (none in next 7 draws)
3 (next on 3rd draw)
*17
7 (none in next 7 draws)
3 (none in next 7 draws)
*19
*19L
*17

That is 20 games to get 8 with guaranteed-mulligan draw that fails to have 0-cost card.  In NONE of those cases did one of the two Towers show up before the 15th card (7 opening hand + 7 mulligan hand).  I'm beginning to think Elements is programmed to simply place the opening fail draw on top of the deck once it mulligan draws!  This is demonstrably different than the Xeno simulator which is proven (above a few posts back) to shuffle after a mulligan draw.

[OldTrees]

That is 20 games to get 8 with guaranteed-mulligan draw that fails to have 0-cost card.  In NONE of those cases did one of the two Towers show up before the 15th card (7 opening hand + 7 mulligan hand).  I'm beginning to think Elements is programmed to simply place the opening fail draw on top of the deck once it mulligan draws!  This is demonstrably different than the Xeno simulator which is proven (above a few posts back) to shuffle after a mulligan draw.

Well I was wrong, I guess Zanz did make another big mistake in the coding. Ouch, and this one is worse than when Chriskang noted and informed Zanz about the RNG not being random(since corrected).

So in conclusion, Faro was right that a failed mulligan is the worst starting hand since you won't get that 1st pillar until the 15th card or later.

[rob77dp]

It also means the Xeno Simulator has a fairly large gap between its results and that of in-game experience.  One gap there is the lack of shards (and salvagers?) in the simulator along with this one of not reshuffling the deck after a mulligan draw (especially notice-able when the mulligan draw still fails)!

[ElementalDearWatson]

I just played my anti Jezebel hand.  It's a 60 card deck, meaning that it's somewhat prone to bad draws, but it's got 20 Towers and 6 Hourglasses, with a mark.  The game lasted 9 rounds, I didn't get a single Hourglass and it was only on the very last round that I got a Tower.  Obviously a failed Mulligan, but at the very least an Hourglass would have been nice.

[Faro]

So in conclusion, Faro was right that a failed mulligan is the worst starting hand since you won't get that 1st pillar until the 15th card or later.

Strange that someone with more than 10000 posts didn't know this...
I was completely sure of that rule, so I'm glad to have start this topic. (Zanz were are you?)