Getting that first turn

Hey there!

Today I am going to touch a topic that is more... universal - it applies to every army you might play.
We are an army made out of paper that sits in vehicles out of paper. We suffer from concentrated fire more than any other army. Going first or second at your will is desirable. In previous editions you kind of relied only on the decision of your opponent in case he deployed his army first. With the changes to pre-game (which make the game more dynamic) there is some tactics involved in list building. The less drops you have, the bigger is the chance for that +1 to your roll.
How likely are you to win the roll-off if you have finished deploying first?

TL;DR its $ \frac{21}{31}=67,7 $ %

How is that? In a normal roll of the chance is 50%. Out of 36 possible rolls you win 15 times, he/she wins 15 times and you have to reroll 6 times. It's 50/50. But when the modificator is involved everything changes. Out of 36 rolls if you have the +1 you win 21 times, he/she wins 10 times and you draw 5 times. Why then 21/31 and not 21/36? Well... with every draw you have to reroll and then you still win 21/36 times but in case of a draw you reroll the 2nd time, and then.... It can theoretically happen MILLLLLLLLION times in a row. To calculate the odds of winning a roll of with one player having +1 to the result you actually need to solve a limit of: $ \lim_{x\to\infty} \frac{21}{36} \sum_{i=0}^{\infty}(\frac{5}{36})^{i} $

The graph shows a chance for a player with a +1 to win a roll of after x number of rerolls (starting with 0 as the original roll).

On the Y axis you have the chance in decimal of winning the roll of, on the X axis is the nth reroll.
The orange line is equal to 21/31. The blue line shows the odds of winning the roll of, after nth reroll, with 0 being the original (first)  roll.

It doesn’t matter what is the difference in number of drops between you and the opponent. Having 2, 4, 10, 20 more drops will still mean you have only 10/31 of winning the roll-off. In ITC you might wanna go first (to secure alpha strike) or second (to have better control over the objectives). If you lose the roll-off, you have still 1/6 chance of seizing the initiative, so that is effectively ~32% chance of going first (assuming your opponent decides to start the game). It is slightly less than 1/3 of your games.

Can this knowledge help us when building our list for a tournament? The answer is yes!
NOTE: If you have the same number of drops then a person that started deploying first gets the +1

I took top16 army lists from adepticon 2018, counted the number of drops they had and based on that info I created a normal distribution graph. In the table below, you can see probability of our opponent having less than X number of drops and of us winning the roll-of.

While building your list you should keep track of how many drops you have. While approaching 19 drops, you will be able to choose whether to go first or second ~~50% of the games. No matter how few drops you have, you will never go over 67,7%, and even while having as few as 8-9 drops you have 2/3 chance.

An example: Hero in one of his posts proposes a full Kabal 2000pts roster, which has at least 11 drops. It means, that the odds of him winning the roll are
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65,02%

Conclusions:
1) Having more drops than our opponent gives us 32% chance of getting first turn
2) You cannot statistically begin more than 67,71% of your games
3) Before you start deploying your army check your opponent’s roster and count his minimal # of drops - if it is higher than yours you can deploy more offensively, if it's lower rather defensively

I hope that this article was interesting, if you have any feedback you can leave it here below, in my thread at The Dark City or in the Dark Eldar Facebook group.

See you next time!