quote:Steve, I guess you're just more visual than me. I just the series summation formula, which is:
ORIGINAL: Shannon V. OKeets
About the Initiative calculations, ...
I always picture a 10 by 10 grid, with the rows being Axis rolls (or Allied) and the columns being the other side's.
Each side rolls a 10 sided die and using the two rolls you identify a cell in the grid.
Now, running down the diagonal are results of when the two rolls are equal. If there are no die roll modifiers, (Initiative track is on zero), then all results below the diagonal are won by one side and all results above the diagonal are won by the other side. There are 45 results above and 45 below. Hence, the player who wins ties wins the diagonal results has a 55% chance of winning the first set of die rolls, the other side has a 45% chance.
When one side is +1 on the initiative track, the diagonal line that defines tie results moves up or down. There are now 9 results in the diagonal. Also, there are 55 results favoring the player with +1 initiative and 36 for the other player. If the player winning ties has the +1, then he will win the first set of rolls 64% (55 + 9) of the time.
And so on.
This visual of the grid lets me calculate the probability of me getting to decide who goes first next turn fairly easily. I hope it helps others.
Sum(i, for 1 to N) = (N+1)*N/2 and account for the advantage and who win ties by adjusting the that formula appropriately. I've gotten to where I can do it in my head, even calculating the chance of winning both rolls, 1 or 2 rolls, etc.
I'll go through how I did my last calculation. Both sides wanted to move first. The axis had a +1 advantage, would win a tie and if necessary (i.e., lost the first roll) would request a re-roll.
So lets look at this from the allied point-of-view:
The allied chance of even winning the first roll requires that they roll a 3 or higher. That is, no way they can with with a roll of 1 or 2. So 3 through 10. For an allied roll of 3, there's only one D10 roll that the axis can make, and that's 1, where the allies win. For an allied D10=3 there's 1 axis rolls, for D10=4 there's 2 axis rolls, ..., for allied D10=10 there's 8 axis rolls.
So using the series summation formula that means there are 9*8/2=36 roll combinations out of 100 (10 allied x 10 axis D10 combos) in which the allies will win the first roll. So there is a probability of 0.36 that the allies will win the first roll.
If they win the first roll then the axis will request a re-roll. In this case the axis would lose their +1 advantage (i.e., reduced by 1 to 0) but still win ties. Again there is no way the allies can win on a roll of 1, but now they have a chance if they roll 2 or higher. For an allied D10=2 there's 1 axis roll (that the allies win), D10=3 there's 2 axis rolls, ..., D10=10 there's 9 rolls. So there's 10*9/2=45 roll combinations out of 100 that the allies will win if a re-roll is necessary. This gives the allies a probability of 0.45 of winning a re-roll (if necessary).
To get the initiative the allies have to win not only the first roll but a reroll. So the probability that this happens is:
Pr(Allies win initiative) = Pr(Allies win 1st roll) * Pr(allies win 2nd roll | win 1st roll) = .36*.45 = 0.162, or 16.2% that the allies will get the initiative.
This (also) means that the axis have a 84.8% chance of winning it.