It is frustating when battle ends with oposite unit having 1/100 HP left just because each round my unit damaged was 33 not 33,3(3) (so it should won battle) but AI unit got lucky 4 winning strikes in a row).
Using the
Combat Odds Calculator, I came up with an example where the attacker does 33 damage per round:
A(ttacker) original str(ength): 8
A(ttacker) generic bonus: 10
D(efender) original str(ength): 3
D(efender) first strikes: 1
That example represents your Combat I War Elephant attacking an enemy Archer with its built-in First Strike and who is defending on flat terrain, not across of a River, and not inside of a City or a Fort.
Here are the values that the Combat Odds Calculator provides:
Combat odds: 99.94%
Retreat odds: 0%
Survival odds: 99.94%
A experience points: 1
D experience points: 5
A modified strength: 8.8
D modified strength: 3
A hit per round: 33 hps
D hit per round: 12 hps
A odds to win a round: 75%
D odds to win a round: 25%
In particular, notice that when you do 33 hps (hitpoints) of damage, the defender is doing 12 hitpoints of damage. So, even if the defender was able to get "lucky 4 winning strikes in a row," and let's say that it also got its First Strike attack for a total of 5 attacks on you in a row, that would still only be 5 * 12 = 60 hitpoints.
In order for the Archer to win, it would have to win ceiling( 100 / 12 ) = ceiling( 8.3 ) = 9 rounds of combat. Assuming that it won its First Strike attack, that would mean that it would have to win an additional 8 rounds, while you'd only have to win 4 rounds.
And, yet, according to the calculator, the odds for the defender to win a given round are only 25%.
So, not only does the Archer have to damage you in 9 rounds before you manage to damage it in 4 rounds, it also only has a 25% chance of winning each given round, with only the First Strike round being (if I understand things correctly) a 50% of doing either 0 damage or one round-worth of damage (12 hitpoints, in this example) to your War Elephant.
If we plug in equal values into the Combat Odds Calculator, say, for two unpromoted Axemen in open-field combat:
A(ttacker) original str(ength): 5
D(efender) original str(ength): 5
then we get these values:
Combat odds: 50%
Retreat odds: 0%
Survival odds: 50%
A experience points: 4
D experience points: 2
A modified strength: 5
D modified strength: 5
A hit per round: 20 hps
D hit per round: 20 hps
A odds to win a round: 50%
D odds to win a round: 50%
As you can see, a "normal" round will deal 20 hitpoints of damage. So, it's only when you're already at a large advantage in the combat when you will deal 33 hitpoints of damage per round.
Note that if we plug in values as the following, basically giving your War Elephant Combat II against the unpromoted Archer with no Promotions and with no defensive bonuses:
A(ttacker) original str(ength): 8
A(ttacker) generic bonus: 20
D(efender) original str(ength): 3
D(efender) first strikes: 1
then we reach the break point of the War Elephant only needing to win 3 combat rounds in order to win the battle:
Combat odds: 100%
Retreat odds: 0%
Survival odds: 100%
A experience points: 1
D experience points: 6
A modified strength: 9.6
D modified strength: 3
A hit per round: 34 hps
D hit per round: 11 hps
A odds to win a round: 76%
D odds to win a round: 24%
since ceiling( 100 / 34 ) = ceiling( 2.94 ) = 3 rounds of combat.
Also, the Archer now needs to win ceiling( 100 / 11 ) = ceiling( 9.09 ) = 10 rounds (or 1 First Strike round and 9 regular rounds) in order to win the combat, with a slightly-lower (24%) chance of winning any given regular round.
Note that if the Archer did win, it would now receive 6 Experience Points instead of just 5 Experience Points; I believe that it was troytheface who posted about Attacko's suggestion of fighting combats with units even at low odds with the hopes of getting good Promotions. Given the right context (say, toss in unpromoted units when assaulting a City early in the game before siege units as your first attackers and where you have a large stack of units to back them up so that you will actually win the battle for the City), this advice is actually quite sage: your unpromoted Archers, Chariots, or Axemen who go in first don't expect to come out of the battle alive, but if they do, you very well might end up with being able to assign those units 2 Promotions on the following turn. Of course, bring enough other units to actually win the battle for the City, otherwise you might lose your wounded units in the AI's counter-attack before your units can promote on the following turn.
Using a boat-based example:
An unpromoted AI Destroyer attacking your defending, unpromoted (or Flanking I) Battleship:
Combat odds: 22.07%
Retreat odds: 0%
Survival odds: 22.07%
A experience points: 5
D experience points: 1
A modified strength: 30
D modified strength: 40
A hit per round: 17 hps
D hit per round: 23 hps
A odds to win a round: 43%
D odds to win a round: 57%
Give your Battleship Combat I:
Combat odds: 11.21%
Retreat odds: 0%
Survival odds: 11.21%
A experience points: 5
D experience points: 1
A modified strength: 30
D modified strength: 44
A hit per round: 16 hps
D hit per round: 24 hps
A odds to win a round: 41%
D odds to win a round: 59%
Give your Battleship Combat II:
Combat odds: 4.13%
Retreat odds: 0%
Survival odds: 4.13%
A experience points: 6
D experience points: 1
A modified strength: 30
D modified strength: 48
A hit per round: 15 hps
D hit per round: 25 hps
A odds to win a round: 38%
D odds to win a round: 62%
Give the attacking Destroyer Combat I and your Battleship Combat III:
Combat odds: 4.77%
Retreat odds: 0%
Survival odds: 4.77%
A experience points: 6
D experience points: 1
A modified strength: 33
D modified strength: 52
A hit per round: 15 hps
D hit per round: 25 hps
A odds to win a round: 39%
D odds to win a round: 61%
Attacker hitting 17 hitpoints per round = ceiling( 5.8 ) = 6 rounds needed to win
Attacker hitting 16 hitpoints per round = ceiling( 6.25 ) = up to 7 rounds needed to win
Attacker hitting 15 hitpoints per round = ceiling( 6.67 ) = also 7 rounds needed to win
Defender hitting 23 hitpoints per round = ceiling( 4.35 ) = 5 rounds needed to win
Defender hitting 24 hitpoints per round = ceiling( 4.17 ) = 5 rounds needed to win
Defender hitting 25 hitpoints per round = ceiling( 4 ) = down to only 4 rounds needed to win
So, promoting to Combat I means that the attacker needs 1 more round to win the battle.
Promoting to Combat II means that the attacker needs 1 more round to win the battle and you need 1 less round to win the battle (relative to you have 0 Promotions or having the Flanking II Promotion*). You promoting to Combat III with the defender being at Combat I just so happens to give similar results in this case, although with the attacker having just slightly higher odds of winning any given round.
If you are defending from the Coast, I believe that you would need to fill in:
D(efender) situational bonus: 10
If my understanding of applying the Coastal defensive bonus is correct, then doing so appears to be the equivalent of an extra Combat Promotion for the defender, according to the odds shown from the Combat Odds Calculator. So, looking at the numbers above, it means that you can have 1 less Combat Promotion and still get the same results if you end your turn on a Coast square.
* Of course, if you are playing multiplayer, Flanking II has a sneaky effect of making your unit immune to First Strikes, so if someone has been making Drill IV Privateers and then upgraded them to Destroyers, Flanking II can be of use, even on defence, as Drill IV gives between 3 and 6 First Strikes. But, AIs don't do that; at most, you'll see Drill II for between 1 and 2 First Strikes.
Let's run those odds (the AI having Drill II), just to see how the numbers look:
Attacking Destroyer has Drill II (1 First Strike chance and 1 First Strike = 1 to 2 First Strikes)
Defending Battleship has Combat II while not being on the Coast:
Combat odds: 6.47%
Retreat odds: 0%
Survival odds: 6.47%
A experience points: 6
D experience points: 1
A modified strength: 30
D modified strength: 48
A hit per round: 15 hps
D hit per round: 25 hps
A odds to win a round: 38%
D odds to win a round: 62%
Attacking Destroyer has Drill II (1 First Strike chance and 1 First Strike = 1 to 2 First Strikes)
Defending Battleship has Flanking II (no combat bonus but we remove the attacker's First Strikes, meaning the same case as our first example of no Promotions on either side) while not being on the Coast:
Combat odds: 22.07%
Retreat odds: 0%
Survival odds: 22.07%
A experience points: 5
D experience points: 1
A modified strength: 30
D modified strength: 40
A hit per round: 17 hps
D hit per round: 23 hps
A odds to win a round: 43 %
D odds to win a round: 57 %
Here, we have to look at the Combat odds, since First Strikes are involved, and we see that Combat II pulls far ahead with the attacker only having a 6.47% chance to win, instead of a 22.07% chance to win when you promoted up to Flanking II.
If the attacking Destroyer has Drill IV and the defending Battleship has Combat II when not on a Coast:
Combat odds: 14.46%
Retreat odds: 0%
Survival odds: 14.46%
A experience points: 6
D experience points: 1
A modified strength: 30
D modified strength: 48
A hit per round: 15 hps
D hit per round: 25 hps
A odds to win a round: 38%
D odds to win a round: 62%
Even in this case, the First Strikes are not enough to overcome the gains associated from hitting the breakpoints of the attacker needing an extra round and the defender needing one less round to win.
So, even in the contrived example, Flanking II isn't doing a lot for you on defence. If there was going to be a lot of stack-versus-stack boat combat in your game, you could argue that Flanking II on a Battleship would allow it to do some Collateral Damage to the entire stack of defenders while having a chance to retreat from the battle. But, in practice, this type of a battle is quite rare and you're more likely to be fighting small skirmishes with small numbers of enemy boats. You're probably most likely to fight a larger number of AI boats in a single turn of combat when on defence when an AI throws boats from multiple squares against your same square. Thus, go with the Combat Promos.
Flanking I is nice for transport ships, though, as, if you are Charismatic with a Drydock, you have a Great General settled in a City with a Drydock, or you can have your Drydock-built boat earn 1 Experience Point, you can get your Galleon or Transport up to 5 Experience Points so that it can promote to Navigation I for +1 movement point, which is a Promotion that gets unlocked by Flanking I. I would not aim to get Navigation I for your combat boats, though; those boats can already move far enough on their own and you want them to have Combat (or maybe Drill) Promotions for doing the actual fighting.