Deck size & Probability - A case for (slightly) bigger deck

[MistahBoweh]

But they don't have the same speed nor reach as mages. A priest tidal waves and they've reset themselves too, a mage novas and keeps on burnin'.

As for mage vs dc, I told you, did I not? You need burn for 40 damage minimum plus 5 cards at least to sac for resources. DC does not have 40 health. Assuming you're not completely allyless, the gain can be pushed through, and you can safely have a set of Shriek of Vengeance slipped in as some cards you'd be saccing in other matchups.

Do you see, though? Maj has a plan to win. Dc has a plan to win that happens to be good vs Maj. DC is advancing his own goals, nothing more. The only counters you should need is to stop enemy win conditions that directly inhibit your own. Say, acid Jet is a good way to stop KP. SS is just stall and can safely be attacked into.

Nobody needs answers to everything in SE. Simple as that. They all just need to counter decks that are natural counters to their own. Otherwise it's an autoloss. Nothing else is an autoloss. Nothing else demands a counter. I fail to see the need everyone has for having to make extra room in their deck for stacks of situational cards that are only ever wanted in one specific matchup.

[Atomzed]

Nobody needs answers to everything in SE. Simple as that. They all just need to counter decks that are natural counters to their own. Otherwise it's an autoloss. Nothing else is an autoloss. Nothing else demands a counter.

I like the principle behind it.

If i interpret it correctly, the principle behind creating a good deck is to ALWAYS identify the winning condition. E.g. portal + gravebone ability + SK, Gwen + SS + rapid fire, Elementalis weenie rush + shard, Logan JD + Dual Wield-smash all, Zhanna KP + Aeon + ally, Eladwen burn-frost-burn.

At this deck creating stage, you should not worry about what others are doing. There's no need to worry abt the meta, bec you need to test out whether the winning conditions work. Some winning conditions are more difficult to achieve than others (mournblade + tidal wave). Others are a lot more consistent and easier to achieve (elad burn-frost-burn).

After you have a great deck with consistent winning conditions, you are on your way to winning. Go conquer the QM!

I fail to see the need everyone has for having to make extra room in their deck for stacks of situational cards that are only ever wanted in one specific matchup.

But the last sentence implies that you got my argument wrong. You are implying that I'm saying to include the few more cards for a "just-in-case" scenario. You are missing the point. My argument is that for CERTAIN cards which is very powerful and game changing, you might want to just include it into your great deck, instead of taking out cards and not affect the dynamics too much. And that CERTAIN cards need not only useful for a particular hero or particular situation.

For example, Resurrection. If i include resurrection in my deck, it will not just be "to counter DC mill". There are other useful situations in which you can use it. For example you managed to get a KP out. You cast resurrection and for the next few turns, you are guarrantee to get allies to take full advantage of the KP. Game changing.

Another example, Eternal Renewal. If your deck is a weenie rush deck, using Soul Reaper for healing, adding 2x ER will have great synergy with your deck. After casting SR, all allies will be removed, so all the other great abilities like Mind Control, Energy Discharge will be left in the deck. Casting Eternal Renewal allows you the chance to recycle those great abilities again, in double quick time. Eternal Renewal is not only included "to counter milling".

Like I said many time in the thread, the probability was done to show that adding 2 cards have some impact on the deck consistency, but it may not be as significant as everyone think it is. I am arguing that for specific cards like Resurrection and Eternal Renewal, you may be better off just adding 2 copies into your deck instead of trying to remove some useful cards in your well-tuned deck.

Also, you will realised that this argument of slightly bigger decks will not be applicable to all heroes. Like mages. If i playing a pure burn mages, I will never go beyond the minimum no. As you said, consistency...and the fact that there's no counter to DD now. If i playing warriors, i will also not go beyond the minimum no. Bloodlust is a great card drawing engine that gives the greatest advantage for a mid-duration game. I will not want the game to go into long-duration, bec the cumulative dmg from Bloodlust is going to be more significant. As such, the chances of me drawing that 2-additional cards is not going to be high.

[EPITAPHEVERMORE]

Doing a thought experiment here. Let's say you have a wonder Elemental deck at 40 cards including hero. Your deck have a great mix of cards, and you tuned it extremely well such that all the cards were necessary. You play 100 games, and you roughly win abt 90 games (90% win rate). Then some genius/joker came up with a wonderful/boring milling deck that caught on within SE community. Suddenly, you see an increase in that particular hero in QM; in 100 games, you see an increase of that hero usage from 5% to 30%. Your deck has 30% of winning such a milling deck, which is terrible cos you are so used to winning 90% of the time. And you realise that Eternal Renewal is a great counter card to the milling deck and will increase your win% to 100% against that milling deck.

What i'm describing is simply the shifting meta. There can be a few responses (not all the possible responses are listed here) to such a scenario
1) You suck it up and just prayed hard that you dont meet the milling deck hero
2) you revise your deck and replaced some cards with ER. What it means in Cruxx "Rule of 6" lingo is that you have reduced the consistency of a particular goal in your deck, and added in another sub-par efficiency goal.
3) You add in ER cards, with out removing any cards from the original deck, increasing your deck size to 42. This is what I'm arguing for.

OR
4) Identify the what type of decks in the old meta have changed to milling decks.(25%) and re-asses your build against the current meta.

eg; maybe there are now less ally heavy decks, so you could potentially reduce board control cards.
even if this means reducing your winning rate from 90% to 65% V's ally heavy decks, if they now only make up for 20% of the meta instead of 40%, you can be rest assured that this is the right decision because
by adding ER you have increased your winning rate from 30% to 100% for milling decks. (30% of the meta.)

In short, no matter how perfect any build may be, it's only perfect against the current meta.
For this reason every deck should be re-assessed against the changing meta on a regular basis.

[Raphael Majere]

On tapatalk, hence cannot type much.

I cite an example. If I play amber for the current meta, I would want to put in 2 enrage. If I don't draw it, it's likely I will lose to elad. Hence, 2 is better than 1. But am I going to cut 1 card just to put 1 extra enrage in?

The simple approach might be simply to insert 1 more enrage.

Cutting 1 card to accommodate 1 enrage makes little difference, why? Cos I might not even draw it in the first place! With only 2 enrage in a 40 card deck, there is a high chance I won't draw it vs elad.

2 enrage in 40 card deck vs a 41 card deck. Marginal difference.

[MistahBoweh]

Not really Raph... It's not just about the odds of drawing one specific card. When you add slots to your deck, you reduce the odds of drawing EVERY OTHER CARD IN YOUR DECK. So you now have less chance of a t1 kris, less chance of a t2 bf or Puwen, less chance of a t3 Jas or Aldon, less chance of a t4 JD. And, yeah, 1-2% here and there may not seem like much. But it adds up when you consider the odds of getting each ideal play each turn of every game. For someone like yourself who plays a lot of games, that's actually a HUGE deal.

[Atomzed]

OR
4) Identify the what type of decks in the old meta have changed to milling decks.(25%) and re-asses your build against the current meta.

eg; maybe there are now less ally heavy decks, so you could potentially reduce board control cards.
even if this means reducing your winning rate from 90% to 65% V's ally heavy decks, if they now only make up for 20% of the meta instead of 40%, you can be rest assured that this is the right decision because
by adding ER you have increased your winning rate from 30% to 100% for milling decks. (30% of the meta.)

In short, no matter how perfect any build may be, it's only perfect against the current meta.
For this reason every deck should be re-assessed against the changing meta on a regular basis.

Agree completely with the point. Which is why tourney winners are usually good readers of the meta. And they make adjustments to their well-tuned deck to maximise their chances of winning the tourney.

Not really Raph... It's not just about the odds of drawing one specific card. When you add slots to your deck, you reduce the odds of drawing EVERY OTHER CARD IN YOUR DECK. So you now have less chance of a t1 kris, less chance of a t2 bf or Puwen, less chance of a t3 Jas or Aldon, less chance of a t4 JD. And, yeah, 1-2% here and there may not seem like much. But it adds up when you consider the odds of getting each ideal play each turn of every game. For someone like yourself who plays a lot of games, that's actually a HUGE deal.

+1

Dnd, that's why i empahasise that the approach is not for ALL cards. It's a calculated risk. It may be worth it for some decks, it may not be worth it for some decks. What the article is trying to argue is that for certain key cards, it may be a good risk to take.

Of course, this risk analysis differs with different ppl. What i'm offering is an analysis of the figures (and i did state that the reduction of % is 2% for each turn), and my interpretation for the probability, and my conclusion that certain cards are worth the risk.

[Atomzed]

(Note: DNDfreak and Ringel pointed out that assumption 7 is wrong. And they proposed a way of estimating the probability which will be closer to the actual probability. See the insert at bottom of post)

Let's do some simulation exercises to check out the impact of increasing deck size on getting the ideal hand.

As the maths can get quite confusing, i going to put out the maths for everyone to check my steps. If i got the maths wrong, pls correct me (and i will get to learn from my mistakes )

Assumptions:-
1) The ideal hand is T1 Kris, T2 Puwen, T3 Jasmine/ Aldon, T4 JD, T5 Raven.
2) No additional card draws from bad santa or blood frenzy (since we just want to check for ideal hand).
3) First player
4) The deck has 4 copies of the cards in the deck.
5) 1 card is sac every turn.
6) The ideal hand need not be drawn on starting hand. It may be draw through card draws.
7) Each turn probability is calculated independently, i.e. at each turn, i assume that the remaining cards are re-drawn. This actually is more lenient bec at each turn, you sorta "reshuffle" your hand and get a chance to re-draw. This assumption is the most controversial, but it remains here purely bec I can't think of a less cumbersome way of calculating the probability. I didn't think it was accurate just to calculate the odds of getting 1x copy of the ideal hand on the starting hand; bec you always have a chance to draw the card you need next turn. So i resorted to doing it in this manner

i'll appreciate it if someone can help me with the last assumption.

Maths for 39 card deck
1) T1 parameters. Hand size = 6, Target card = 4, Total Deck size = 39. Prob of getting at least 0 Kris = 49.7%.
So prob of getting at least 1 kris = 100-49.7% = 50.2%

2) T2 parameters. Hand size = 5, Target card = 4, Total Deck size = 37. Prob of getting at least 0 Puwen = 54.4%.
So prob of getting at least 1 Puwen = 100-54.4% = 45.5%

(note hand size is reduced bec every turn you have a net loss of 1 card (you draw 1 card, sac 1 card and play a card. Deck size is reduced bec of assumption 7; i have to reduced the deck size each turn, bec Kris was played and 1 card was sac the previous turn.)

3) T3 parameters. Hand size = 4, Target card = 8, Total Deck size = 35. Prob of getting at least 0 Jasmine/Aldon = 33.5%.
So prob of getting at least 1 Jasmine/Aldon = 100-33.5% = 66.4%

4) T4 parameters. Hand size = 3, Target card = 4, Total Deck size = 33. Prob of getting at least 0 JD = 66.9%.
So prob of getting at least 1 JD = 100-66.9% = 33.0%

5) T5 parameters. Hand size = 2, Target card = 4, Total Deck size = 31. Prob of getting at least 0 Raven = 75.4%.
So prob of getting at least 1 Raven = 100-75.4% = 24.5%

So estimated prob of getting the ideal hand (be it from starting hand or from draw) = 1.23%

-------------
now lets repeat the same steps, but with a 41 deck size.

Maths for 41 card deck
1) T1 parameters. Hand size = 6, Target card = 4, Total Deck size = 41. Prob of getting at least 0 Kris = 51.7%.
So prob of getting at least 1 kris = 100-51.7% = 48.2%

2) T2 parameters. Hand size = 5, Target card = 4, Total Deck size = 39. Prob of getting at least 0 Puwen = 56.8%.
So prob of getting at least 1 Puwen = 100-56.8% = 43.6%

(note hand size is reduced bec every turn you have a net loss of 1 card (you draw 1 card, sac 1 card and play a card. Deck size is reduced bec of assumption 7; i have to reduced the deck size each turn, bec Kris was played and 1 card was sac the previous turn.)

3) T3 parameters. Hand size = 4, Target card = 8, Total Deck size = 37. Prob of getting at least 0 Jasmine/Aldon = 35.9%.
So prob of getting at least 1 Jasmine/Aldon = 100-35.9% = 64.1%

4) T4 parameters. Hand size = 3, Target card = 4, Total Deck size = 35. Prob of getting at least 0 JD = 68.8%.
So prob of getting at least 1 JD = 100-68.8% = 31.3%

5) T5 parameters. Hand size = 2, Target card = 4, Total Deck size = 33. Prob of getting at least 0 Raven = 76.8%.
So prob of getting at least 1 Raven = 100-76.8% = 23.1%

So estimated prob of getting the ideal hand (be it from starting hand or from draw) = 0.97%

TL;DR
The ideal hand % for 39-card deck is 1.23%
The ideal hand % for 41-card deck is 0.97%.

You can look at the 2 figures and have 2 possible interpretations
a) "Crap, having 2 more cards reduce my % of getting ideal hand by 0.25%. Which equates to 0.79 times less likely to get more ideal hand (0.97/1.23). I'm not going to add that 2 cards in my deck."

b) "Crap, the ideal hand % is already so low at 1.23%. I'm very unlikely to rely on the ideal hand to win, so i might as well add in that 2 additional enrage that i need, but could never find space to include it in."

The stats are shown clearly. You make the decision of whether to add that 2 cards in.

----
(as DNDfreak and Ringel correctly pointed out, assumption 7 is wrong, so the estimated probability is incorrect. A simple and close-enough estimates would be Ringel calculations)

I agree, assumption 7 is too strong. For a quick estimate I would use the hypergeometric calculator, but reduce deck size by 1 each time, and keep draw fixed at 6 cards (You draw 1 new card but put aside the card you needed)

P(picking one of 4 Kris in 6 cards in a 39 card deck)*P(picking one of 4 Puwen in 6 cards in a 38 card deck)*P(picking one of 8 Aldon/Jasmine in 6 cards from a 37 card deck)*P(picking one of 4 Jewelers in 6 cards in a 36 card deck)*P(picking one of 4 Ravens in 6 cards in a 35 card deck)

Repeat, starting at 41 cards.

It wont be right, but I think it will be a better eyeball estimate.

Eyeball estimates: 39 cards: Perfect hand, probability 0.0599
41 cards: Perfect hand, probability 0.0495

[Raphael Majere]

Not really Raph... It's not just about the odds of drawing one specific card. When you add slots to your deck, you reduce the odds of drawing EVERY OTHER CARD IN YOUR DECK. So you now have less chance of a t1 kris, less chance of a t2 bf or Puwen, less chance of a t3 Jas or Aldon, less chance of a t4 JD. And, yeah, 1-2% here and there may not seem like much. But it adds up when you consider the odds of getting each ideal play each turn of every game. For someone like yourself who plays a lot of games, that's actually a HUGE deal.

Which is why I was citing an example and highlighting a case.

Visualise an example: Whereby the T1 Kris, T2 BF/Puwen, T3 Jas/Aldon is not crucial for victory and you'll see why it's 'viable' to go beyond 40.

TBH, I voiced my disagreements with Atomzed's theories on this topic at A1.

Based on your example, I actually have no issues playing at 41 or 42.

1. At T2, I can play Puwen/BF/BS. (12 cards)
2. At T3, I can play Jas/Aldon (8 cards) - or any of the t2 cards.
3. I hate doing T1 Kris most of the time anyways. I usually don't.

While we can probably discuss probabilities until the cows come home, luck plays a huge factor in the game too. Esp. When the game is reduced to top-decking.

When it reaches that point, sometimes you wish you had place that 1 x ER in. Or 1 x KP. Or 1 x SR. If you did not have it in the deck in the first place, of course you'll NEVER draw it.

I've played games that I was leading and winning all the way and BAM! SS Armor comes out - topdecked and gained that extra few turns/cards for Elad and she killed me. OR my opponent's Logan drew a KP and his Medusil killed my PB and swung the game. The medusil gain 1 life and my GB cannot FB the medusil to death.

I also argue that a better 'player' can reduce/manage the 'bad' probabilities better than a average player - due to a more acute sense on what to sac.

I also want to highlight Crispy's Amber deck - 47 or 49 cards. And he went to finals of Crown Tourney. (Eventual Winner: Sooth, who did lose a game vs Crispy)

Crispy's Amber beat Master Blakes and Hatts's 40 card Elad as well, on his way up.

Still on the topic of Amber: suppose that BF is important to her. (I must highlight that some A1 members plays with only 3 BF in some warrior 1.27 decks). Drawing 6 cards out of 40 vs Drawing 6 cards out of 41 has hardly any difference in trying to get a BF by T2.

[Padawan Pete]

I am with raph on this on, except whst I do is insert 2 cards of similar value eg: 1 enrage and 1 rampage or 1 kp depending on what I build against, as a very loose example.

[kamman13]

But they don't have the same speed nor reach as mages. A priest tidal waves and they've reset themselves too, a mage novas and keeps on burnin'.

As for mage vs dc, I told you, did I not? You need burn for 40 damage minimum plus 5 cards at least to sac for resources. DC does not have 40 health. Assuming you're not completely allyless, the gain can be pushed through, and you can safely have a set of Shriek of Vengeance slipped in as some cards you'd be saccing in other matchups.

Do you see, though? Maj has a plan to win. Dc has a plan to win that happens to be good vs Maj. DC is advancing his own goals, nothing more. The only counters you should need is to stop enemy win conditions that directly inhibit your own. Say, acid Jet is a good way to stop KP. SS is just stall and can safely be attacked into.

Nobody needs answers to everything in SE. Simple as that. They all just need to counter decks that are natural counters to their own. Otherwise it's an autoloss. Nothing else is an autoloss. Nothing else demands a counter. I fail to see the need everyone has for having to make extra room in their deck for stacks of situational cards that are only ever wanted in one specific matchup.

Well, I guess I do view things differently than you. I don't want to be able to exactly counter every hero with every deck, but I do want greater than 40% win records against every dominant hero in the meta. Not having an auto-loss isn't good enough for me, I want near 50% win ratios against every potential hero. What I basically find is that by going over 40 cards, I can include the counters to bump my win record to about 50% against all dominant heroes, at the cost of reducing my win record from say 90%-80% against the heroes I'm naturally good against (because my draw is less consistent. So less consistency at my deck strategy, vs. more consistent against various match-ups. This seems to be what you are also saying in your last paragraph, which I completely agree with. our disagreement seems to be that I find that to "counter my nature counters," I often need to go above 40 cards, whereas you seem to think this should never be needed. And IMO, a stack of cards that counter one of the dominant heroes in the meta is hardly situational.

Also, adding in a few SoV is not enough for Maj to counter DC. Imagine the perfect majiya draw, with her going first:
T2 - Bazaar
T3 - FB
T4 - LS
T5 - Nova
T6 - 2x FB
T7 - FB, LS

This equates to 1 damage shy of defeating DC (27 damage). Now what is DC's best response (and this isn't nearly as improbable as Majiya's draw)?
T3 - WBT
T4 - Lone wolf
T5 - SpeedStrike
...
then after Majiya's T7, he uses his ability 2x in a row, and SF for a third time. This equates to 18 damage to Majiya on these 3 turns, plus another 7 or so damage to Majiya from attacking pre-T7, plus the 5 damage from the nova. Plus he heals 10 damage from Lone wolf in this time, more if he carries reg. This effectively gives DC more than 40 health. Even if Majiya shrieks his WBT or Lone wolf instead of waiting for the FM, she loses.

To counter (i.e. not lose 90% of matches against) DC, Majiya needs several shrieks, allies, and maybe a Snow Sapphire or two. This has now become a significant portion of a 30-card deck, probably around 8-9 cards. Just to not auto-lose to one hero. A 35 card Majiya, though, would probably fare much better against DC, though with slightly reduced win percentages against other heroes.