61.8% Chance to Go First with +1.

[Diagramdude]

If you have fewer drops than your opponent and get the +1 to your roll to go first, you have a 61.8% chance to win first turn.

 

  1 2 3 4 5 6

1 L T W W W W 

2 L L T W W W

3 L L L T W W

4 L L L L T W

5 L L L L L T

6 L L L L L L

 

Based on this table we can see the player with the +1 advantage will win 21/36 rolls, tie 5/36, and lose 10/36. It gets tricky with the rerolling of ties, so I simulated one million rolls:

 

Roll 1:

583333 138889 277778

 

Roll 2 takes the 138,889 ties and puts them through the same probabilities:

81019 19290 38581

 

And so on:

Roll 3

11253 2679 5358

 

Roll 4 1563 372 744

 

 

Roll 5 217 52 103

 

I stopped after 5 rolls because there were only 52 ties out of a million. So after 5 rolls, the player with the advantage wins 677,384 and loses 322,564. All that remains is the Seize roll.

 

For the 677,384 wins, one sixth of them will be losses due to Seize. So of the 677,384 winning rolls, 112,897 will get Seized on. However, of the 322,564 losses, the advantage player will Seize 53,761 of them. 

 

So out of one million games, ignoring the 52 ties still remaining after 5 iterations, the final amount of wins for the advantage player will be 677,384 + 53,761 - 112,897 = 618,248 wins

 

And the losses will be 322,564 + 112,897 - 53,761 = 381702 losses. 

 

So the player with the +1 advantage has a 61.8% chance to go first. 

 

[9x19 Parabellum]

We need a new Imperial faction; Ordo Arithmatica, and Diagramdude shall be the founding chapter master.

[brother_b]

All hail the Diagramnassiah!

[Karhedron]

But then your opponent has a 16.67% chance to go first if he siezes the initiative. So if you take that into account, you only have a 51.6% chance of going first, even if you finish deploying first. This means the benefit of reducing your deployment count is minimal.

[Diagramdude]

Karhedronuk, no, I accounted for the Seize. 
 

For the 677,384 wins, one sixth of them will be losses due to Seize. So of the 677,384 winning rolls, 112,897 will get Seized on. However, of the 322,564 losses, the advantage player will Seize 53,761 of them. 

 

So out of one million games, ignoring the 52 ties still remaining after 5 iterations, the final amount of wins for the advantage player will be 677,384 + 53,761 - 112,897 = 618,248 wins

 

And the losses will be 322,564 + 112,897 - 53,761 = 381702 losses. 

 

So the player with the +1 advantage has a 61.8% chance to go first. 

[9x19 Parabellum]

Karhedronuk, did you really just question the Chapter Master of the Ordo Arithmatica?  Bad move, bro. 

[Karhedron]

Karhedronuk, did you really just question the Chapter Master of the Ordo Arithmatica?  Bad move, bro.

My humble apologies. I shall report for one hour of penitential calculus at my nearest Officio Arithmatica.

[9x19 Parabellum]

 

Karhedronuk, did you really just question the Chapter Master of the Ordo Arithmatica?  Bad move, bro.

My humble apologies. I shall report for one hour of penitential calculus at my nearest Officio Arithmatica.

 

 

Penitential Calculus.   Now THAT is a punishment, if I ever heard of one.

[Halandaar]

Is this fundamentally different from the thread you started in this forum 2 weeks ago where a figure of around 60% was already arrived at?

[Panzer]

Is this fundamentally different from the thread you started in this forum 2 weeks ago where a figure of around 60% was already arrived at?

I knew we had such a thread already! lol

[Diagramdude]

Yes I should have just continued in that thread. In that thread though the 60% was a guesstimate, this thread is, what I believe, is the actual math.

[Arkhanist]

I believe there is a simpler way to calculate this - we ignore ties. We don't care how many times a tie is rolled, as it has no effect on the re-roll to break it - we throw away the original result, just as if we started over due to a cocked roll.

 

So, we take the original odds table, throw away the 5 ties, thus going from ... in 36 to .... in 31. That means 21/31 rolls will win, and 10/31 will lose (when you have a +1). We then factor in the seize chance - 5/6 out of times you won't be seized when you win the roll off, and 1/6 times you will seize when you lose the roll off.

 

So that makes the final winning chance as the player with the +1 = (21/31 * 5/6) + (10/31 * 1/6)

Applying rules for fraction multiplication, that becomes

 

win chance = (105/186) + (10/186)  = 115/186.

 

115/186 ~= 0.618279

 

So your odds of going first with a +1, including the seize roll is 115/186, or ~ 61.8%.

 

Which is close to the result gained experimentally with a simulated million rolls (0.618248), which is nice

Edited February 27, 2018 by Arkhanist

[Montoya]

Arkhanist has it right.

 

If you roll a tie, it basically just nullifies the roll as it has no impact on the subsequent roll that you need to make (it is as if the first roll never happened.)

[Diagramdude]

Elegant solution! The end results agree. I would say a 61.8% chance to go first is a significant incentive to influence list building.

[Calistarius]

For sure, if you’re playing standard mission rules. Other formats favor going second. Especially, if the format uses progressive objective scoring. I’m likely in the minority here, but I like to see what my opponent is doing and then respond with a hammer blow.

 

It’s cool to see this from a math standpoint though!

[Xhris the Unpronounceable]

You can’t argue with math. Well you can, but you’d be wrong...