Probability of Ennead doing damage on first turn

Other Games SotM Game Engine
1

Recalling a five hero game against the Ennead in which the only damage my heroes took was 3 points of damage self-infliced by Legacy's Heroic Interception, I wondered what are the chances that the Ennead wouldn't do damage on their first turn in a five hero game?

 

Of the 25 Ennead non-character cards, only eight don't immediately deal damage: Two each of Rise To Power, Taste of Immortality, The Desert's Wrath, and The Grave Beckons.

Rise To Power plays another card, so from a probability standpoint, what it's really doing is decreasing the pool of cards to consider, from 25 to 23.

Taste of Immortality neither does damage nor triggers any of the three symbols: Green, Purple, or Red, so that's 2/23 chance of no damage.

The Desert's Wrath has a green symbol. There are five members of the Ennead that will deal damage if triggered by a green symbol, so there is no way to choose five members of the total nine that WON'T deal damage, so The Desert's Wrath will cause at least some damage to be done.

The Grave Beckons has a purple symbol. There are three members of the Ennead that will deal damage if triggered by a purple symbol, so it's possible for the starting members of the Ennead to not deal damage if triggered by a purple symbol: 6/9 * 5/8 * 4/7 * 3/6 * 2/5 = 1/21. The total probability of The Grave Beckons not leading to damage is therefore 2/23 * 1/21.

Adding these together gives 2/23 + (2/23 * 1/21) = 2/23 * (1 + 1/21) = 2/23 * 22/21 = 9.1%.

End result? Nearly a ten percent chance of the Ennead not dealing damage on their first turn in a five hero game. Quite surprising!

2

That's much higher than I would have expected.

3

A couple of notes:

  • Don't forget that, if Atum is in play, the Grave Beckons will cause him to search for a red card. You might as well treat Atum as dealing damage off of a purple card play, since all of the red cards deal damage.
  • If we want to get pedantic, Osiris triggering off a purple card doesn't cause the Ennead to deal any damage - the heroes damage themselves. You may not want to treat Osiris as triggering off of purple cards.

So, theoretically, these two changes could cancel each other out.

4

I worded the situation incorrectly -- I was looking for situations where no damage is done to the heroes, regardless of who does the damage.

I included Atum's play of a red card as doing damage. The three members who cause damage to be done when triggered by purple are Atum, Osiris, and Isis.

5

Are you factoring in that Rise to Power can result in having 1 or 2 more members of the Ennead in play before a symbol card is played?

Also, how about Challenge Mode ;)

6

Hmm ... no, didn't factor in that Rise to Power might bring in more members of the Ennead. Interesting.

It would decrease the percentage by a relatively small amount. In 2/25 cases, the percentage of no-damage scenarios for The Grave Beckons would be less than 1/21, so the already small 2/23 * 1/21 would be reduced by less than ten percent, so no more than .05 percent change. With rounding, that _might_ drop the result from 9.1% to 9.0%. Still, interesting.

 

As for Challenge Mode, I haven't payed close attention and don't know what it is! To the wiki! Oh, wow. Every one-shot has all three symbols. Well that would make the math easier: 0%.

 

Just realized how well The Ennead work with America's Cleverest Legacy.

7

I tried it ten thousand times as an experimental check. Looks like you are correct!

  • Total Damage: 154281
  • Average Damage: 15.4281
  • Zero Damage: 951 times
  • Zero Damage: 9.51 % of the time
  • Time: 641797 ms
8

So cool!

9

Hmmm ... with that many runs, I'm surprised that my calculation isn't closer to the experiment. I wonder if I missed something.

10

Did you run the experiment only on 5-hero teams?

11

Yep! Here's the code, all it does is start the game and play an Ennead card.

 

int someDamage = 0; int totalDamage = 0; int tries = 10000; for (int i = 0; i < tries; i++) { SetupGameController("TheEnnead", "Legacy", "Bunker", "TheWraith", "Fanatic", "Ra", "InsulaPrimalis"); StartGame(); GoToPlayCardPhase(ennead); PlayTopCard(ennead); GoToEndOfTurn(ennead); int damage = 0; foreach (var hero in GameController.Game.HeroTurnTakers) { damage += (hero.CharacterCard.MaximumHitPoints.Value - hero.CharacterCard.HitPoints.Value); } Console.WriteLine("Damage: " + damage); totalDamage += damage; if (damage > 0) { someDamage += 1; } }
12

10,000 tries is not a very big sample, btw. The experimental result there does not indicate your answer is incorrect.

13

Having Ra on the team skews the results a little, because then The Desert's Wrath deals damage.

14

Good point! I ran the experiment again Tachyon instead of Ra:

  • Total Damage: 141815
  • Average Damage: 14.1815
  • Zero Damage: 900 times
  • Zero Damage: 9 % of the time
  • Time: 663407 ms
15

 

OOOoooOOOhhh! Good catch!

16

A lot closer. I calculate a 9.08% chance of no damage (conservatively assuming that any additional members of the Ennead would cause some damage to be dealt)

 

Now, if only I knew how to determine how likely it would be to deviate from the expected calculation. Darn it, I took a Statistics class and wanted to learn it and got a decent grade, but still don't really get it.

17

IIRC from my college days, you write the numbers down in a linear fashion, then draw a camel hump over them. Slice it up into, oh, say 5 or 7 pieces, and those sections are your deviations. It's not an exact science mind you, but it sure looks close.

18

Wait a sec. Including Ra makes it more likely that damage will be dealt, so replacing Ra should make is LESS likely that damage will be dealt, meaning that the percentage time that damage isn't dealt should go UP, not down ...

19

It's a small sample size. 10,000 seems like a lot, but it's not.

20

All I know from this is that I want to have access to these functions at MY job.