Hit Probability Table

Discuss strategies of warbarons

Hit Probability Table

Postby KGB » Sat Jul 19, 2014 4:43 am

As most of you know a Pikeman has a strength of 15 and a Lt Infantry has a strength of 10. Thus the Pikeman is 50% stronger than the Lt Infantry and 2 Pikeman should equal 3 Lt Infantry, right? Wrong. The Combat rules are a little more complicated than that making it harder to evaluate the relative value of 2 different strengths when calculating the hit probability.

So below I present the hit probability table for Warbarons at various strengths:


Code: Select all
  |  05 |  10 |  15 |  20 |  25 |  30 |  35 |  40 |  45 |  50 |  55 |  60 |  65 |  70 |  75 |
--------------------------------------------------------------------------------------------
05| 1.00|  .47|  .30|  .21|  .16|  .12|  .10|  .08|  .06|  .05|  .04|  .04|  .03|  .02|  .02|
10| 2.11| 1.00|  .63|  .44|  .33|  .26|  .21|  .17|  .14|  .11|  .09|  .07|  .06|  .05|  .04|
15| 3.35| 1.59| 1.00|  .71|  .53|  .41|  .33|  .26|  .22|  .18|  .14|  .12|  .10|  .08|  .06|
20| 4.75| 2.25| 1.42| 1.00|  .75|  .58|  .46|  .38|  .31|  .25|  .20|  .17|  .13|  .11|  .08|
25| 6.33| 3.00| 1.89| 1.33| 1.00|  .78|  .62|  .50|  .41|  .33|  .27|  .22|  .18|  .14|  .11|
30| 8.14| 3.86| 2.43| 1.71| 1.29| 1.00|  .80|  .64|  .52|  .43|  .35|  .29|  .23|  .18|  .14|
35|10.23| 4.85| 3.05| 2.15| 1.62| 1.26| 1.00|  .81|  .66|  .54|  .44|  .36|  .29|  .23|  .18|
40|12.67| 6.00| 3.78| 2.67| 2.00| 1.56| 1.24| 1.00|  .81|  .67|  .55|  .44|  .36|  .29|  .22|
45|15.55| 7.36| 4.64| 3.27| 2.45| 1.91| 1.52| 1.23| 1.00|  .82|  .67|  .55|  .44|  .35|  .27|
50|19.00| 9.00| 5.67| 4.00| 3.00| 2.33| 1.86| 1.50| 1.22| 1.00|  .82|  .67|  .54|  .43|  .33|
55|23.22|11.00| 6.93| 4.89| 3.67| 2.85| 2.27| 1.83| 1.49| 1.22| 1.00|  .81|  .66|  .52|  .41|
60|28.50|13.50| 8.50| 6.00| 4.50| 3.50| 2.79| 2.25| 1.83| 1.50| 1.23| 1.00|  .81|  .64|  .50|
65|35.29|16.71|10.52| 7.43| 5.57| 4.33| 3.45| 2.79| 2.27| 1.86| 1.52| 1.24| 1.00|  .80|  .62|
70|44.33|21.00|13.22| 9.33| 7.00| 5.44| 4.33| 3.50| 2.85| 2.33| 1.91| 1.56| 1.26| 1.00|  .78|
75|57.00|27.00|17.00|12.00| 9.00| 7.00| 5.57| 4.50| 3.67| 3.00| 2.45| 2.00| 1.62| 1.29| 1.00|



Reading the table you can see the Pikeman (str 15 using row 3) is 59% more likely to get a hit than the Lt Infantry (str 10, column 2). You can say this tiny difference (59% vs 50%) isn't really noticeable. But take a look further down the table at 45 to 30 strengths (another 3/2 ratio). A unit of str 45 is really 1.91 times as likely to get a hit than a unit of str 30, that's almost two times as likely meaning now 1 Pikeman equals 2 Lt Infantry! A unit of str 75 is 3x times as likely to get a hit on a unit of str 50, not 1.5x as you'd expect so now 1 Pikeman equals 3 Lt Infantry.

So what does this mean? Well take a look at a 75 str Barb vs a 5 str unit (Crow/Scout). The Barb is 57x as likely to get a hit. That means on average the Barb hits the Scout 57 times before it takes a single hit. That equates to killing 57/2=26 Scouts per hit! That's what I like to call Rambo Mode :lol: Even against a 20 strength unit (Hv Inf defending) the Barb will hit 12x before taking 1 hit which equates to killing 6 Hv Inf per hit. Finding Ragnarok for example at L1 takes a Barb to 40 str so that vs a 25 strength unit he is now 2x as likely to get a hit vs 1.29.

Notes:
1) If you are wondering why a 3/2 ratio changes so much from 15/10 (1.59) to 45/30 (1.91) to 75/50 (3.0) it's because of the fact that the closer your strength gets to 100 (max combat number) the better your ratios get. That's because at 100 you would be invincible even to a 99 strength unit. In the opposite manner the closer your strength gets to 0 the worse your ratios get because at 0 you would never get a hit on even a 1 strength unit.
2) The numbers in the table represent the (chance to hit for unit A / chance to hit for unit B). So opposite squares multiplied together approximately = 1.0 (total hit probability for that strength combination). For example the 25/5 (6.33) x 5/25 (.16) = 1.01.

KGB

P.S. Apologies for the small font in the table but that's the only way I can get it to fit.
KGB
 
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Re: Hit Probability Table

Postby gil » Sat Jul 19, 2014 10:00 pm

nice table thanks for sharing

this is a discovery of one more unbalanced barbarian attribute
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Re: Hit Probability Table

Postby KGB » Sun Jul 20, 2014 6:20 pm

Barb can be stopped easily with any/all of Orcs/Wolfriders/Ghosts/Assassin hero.

I always always always have some of these units around that reason.

KGB
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Re: Hit Probability Table

Postby gil » Sun Jul 20, 2014 9:46 pm

lets c if i understand here is a list of unfair barb attributes
1. a barb is best for abuse of the 90% rule making a sure win with minimum units lost
2. 1 maxed barb will have more power then an 8 man stack with a maxed pala
3. barb is unaffected by archon
4. barb can keep upgrading his power till level 10 while other heros are maxed at 5
5. barb have more ruin items then all other heros
6. barb can use his items (life,ward) unmaxed while other heros items max at -15 or +30
7. barb will recive more personal power then an entire 8 man stack with pala or DK 8. every level he upgrades
9. barb will be stronger then a 4 turn unit at lvl 2 all other heros will need to be lvl 3 for them to become stronger then a 3 turn unit


and your saying X amount of Orcs/Wolfriders/Ghosts/Assassin that sit around just in case a barb show up will make this fair?
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Re: Hit Probability Table

Postby KGB » Mon Jul 21, 2014 1:42 am

2) Not true. A maxed Paladin gives +11 bonus to his stack. An 8 stack of say Giants with a Paladin will fight at 36 strength. That stack will have about a 50/50 shot a beating a maxed Barb (65 str, 5 hits). Better units that Giants give the Paladin stack an even better chance of winning.
3) Assassins are immune Archons too. Other heroes like Horselord and Ranger barely even notice an Archon because their bonus is SO high that losing some of it doesn't matter much.
4) Not true. The DK doesn't max out at 18 Chaos until L10. The Assassin doesn't max out until L11 (72 ambush, 7 Chaos) The Ranger doesn't until L9 (max in Woods/Swamp/Warding). Horselord doesn't until L9 (Snow/Open/Desert). Valk doesn't max out at 18 Leadership until L6. Even the Paladin can still add Warding once he reaches L5. It's a BENEFIT to max out at L5, not a handicap.
5) Not sure what you mean by this. Any hero can use any item. There are no special items just for a Barb. In fact the Barb can't really search ruins at all due to a VERY low Undead Lore score so Barbs hardly ever have items unless another hero gives them to the Barb.
6) Not true. Other heroes can use Life/Ward items unmaxed just like the Barb. The +30/-15 limit can be exceeded. For example if you have +40 Leadership and face an Archon you lose the Leadership from the Archon BEFORE the cap is applied. Also if your +40 Leadership hero faces a +10 Chaos hero you still have +30 Leadership, not +20.
7) Maybe. Depends on the stack with the Paladin/DK. The Paladin upgrades at every level too. The DK at every other level.
9) So? That's the Barb's only purpose. He does nothing for his own stack of units. If he wasn't stronger than single L4 units he would be useless. A L2 Barb is not going to beat a Spider or Gryphon or Green Dragon at L2 unless he is quite lucky.

KGB
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Re: Hit Probability Table

Postby tabanli » Mon Jul 21, 2014 4:37 pm

A while ago I presented a concept "POWER" which gives the same results as this table. It is based on hit to miss ration. This is the concept: Regardless of the defender, the probability of an attacking unit to hit is proportional to its own strength and probability of avoiding being hit is inversely proportional to (100-Strength) which is the percent to miss. Same thing for the defender. So to find the relative ratio one needs to compare the individual powers.

Power=Strength/(100-Strength)

Power ratio of units represents the ratio of probability of hit.

So let's sat 75 strength is against 25 strength
P (75) = (75/25) = 3
P (25) = (25/75) = 1/3

So 75 strength is 9 times more likely to hit, which is the same as the table (75/25)

Power 75 versus Power 20 is 3 versus 0.25 so the relative hit ratio is 12 which is the same as the table.

Here is the power list.

5 0.05
10 0.11
15 0.18
20 0.25
25 0.33
30 0.43
35 0.54
40 0.67
45 0.82
50 1.00
55 1.22
60 1.50
65 1.86
70 2.33
75 3.00

It is possible to calculate total power of a stack by multiplying the individual powers with individual hits and adding up. It gives an approximate evaluation between the two stacks. This excludes ambush, warding and anti-air abilities.
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Re: Hit Probability Table

Postby tabanli » Mon Jul 21, 2014 4:53 pm

Also for UL POWER TABLE for L1 Ruin

Undead lore 12 -> Hit to miss ratio = 12 / 0 = infinity. You never lose.
Undead lore 11 -> Hit to miss ratio = 11/1 = 11. With 11 undead lore your power is 11.
Undead lore 10 -> Hit to miss ratio = 10/2 = 5. Here comes the dramatic drop
Undead lore 9 -> Hit to miss ratio = 9/3 = 3
Undead lore 8 -> Hit to miss ratio = 8/4 = 2
Undead lore 7 -> Hit to miss ratio = 7/5 = 1.4
Undead lore 6 -> Hit to miss ratio = 6/6 = 1
Undead lore 5 -> Hit to miss ratio = 5/7 = 0.7
Undead lore 4 -> Hit to miss ratio = 4/8 = 0.5
Undead lore 3 -> Hit to miss ratio = 3/9 = 0.33

As you see, regardless of what is at the ruin, your relative ratio of hitting more than doubles when you go from 10UL to 11UL.

UL 11 versus UL 3: UL11 is 33 times more likely to hit
UL 10 versus UL 3: UL10 is 15 times more likely to hit
UL 9 versus UL 3: UL9 is 9 times more likely to hit. So you have 3-4 % chance to get slain here where it is less than 1 per thousand with UL11.
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Re: Hit Probability Table

Postby KGB » Mon Jul 21, 2014 9:07 pm

Tabanli,

Remember it's possible to face Ghouls (5 UL) in L1 ruins. You only showed Skeletons (3 UL).

I remember your Power rankings. Not sure how many understood what you wrote or what it meant when you published it. What's important for players to understand is that the hit probability (Power) changes dramatically as strengths change even though the relative ratios (15/10, 45/30, 75/50) between 2 strengths don't. That's not easy for newbies to grasp as it's not intuitive.

KGB
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