More Bomb Probability Statistics

[OncoByte]

[Update: The results and conclusions in this post do not paint the whole picture. Sanzo's hypothesis is still partially correct. Please see OncoByte's post at the bottom of this page. ~RM]

...in the sense that it will usually hurt a player with many pieces, and leave a losing player mostly unscathed.

but I guess regular bombs already does some of that.

I don't know, jimmi.. Bombs have more chance to hit you if you have a lot of tori, but if you have very few tori each hit you take does a lot more damage.

Onco or someone else may help with some accurate calculation, but I think that the chance of having 25% of you army destroyed by bombs is more or less the same whether (sic) you have 20 tori or 4. Am I wrong?

The question: What are the odds that you will hit 25% of your opponents squad with a volley of Bombs?

The hypothesis: The odds are roughly the same no matter how many tori your opponent has left.

The method: Write an excel macro that steps through all of the different combinations in which a volley of Bombs can hit a given number of enemy tori - calculate the probability of each event - add those probabilities to find the probability of hitting a set number of tori with a volley of bombs. The program allows you vary the number of arena tiles, bombs in a volley, enemy tori, friendly tori, destroyed foes, and Smart status.

The data: Displayed as chance to hit 25% or more of the enemy squad. The effect with Smart Bombs cases are calculated assuming your squad is at full force (20 tori).

Code: Select all

Enemies  Hit   Bombs   Smarties
   20     5+   27.8%   54.4%
   16     4+   32.4%   56.0%
   12     3+   37.8%   58.0%
    8     2+   44.9%   61.0%
    4     1+   56.0%   66.8%

The conclusion: Sanzo's hypothesis is incorrect. You have a higher chance of losing a quarter or more of your squad during a standard 16-bombs volley in a 8x10 arena if you have fewer tori. This difference is less with Smart Bombs, but the absolute chance of losing a quarter or more of your squad is greater with Smart Bombs.

The interpretation: Bombs have a proportionally greater effect on players with fewer tori. They still serve as potential net equalizers, but carry a significant risk of self-obliteration (18.2% if you have a lone torus to start with).

Analysis of the Smart Bombs scenario reveals that Smart Bombs do, in fact, suck. Yes, they could serve as equalizers for the losing player, but by definition, the losing player is less likely to get Smart Bombs since they are losing. Even if they get Smart Bombs, the damage they could do to an opponent is proportionately much less than the damage enemy Smart Bombs could do to them. If you have few tori, the statistical advantage gained by reducing the effective tile numbers goes away and Smart Bombs approach being Bombs without the chance of self destruction.

Future Analysis: Let me know what other situations you want statistics for.

[Rydash]

Wow. The analysis of another situation by Onco that contradicts my thinking. Not hard to contradict my thinking, that is, but hey.

Although, I have an odd situation that I'd like to see...

16 Bomb volley, 8x10 board, 20 vs. 1, and obviously Smart Bombs probabilities need not be applied.

What are the chances that you obliterate a significant amount of opponent pieces (Not sure of a good # for this....a quarter? Half?) compared to the likelihood that you wipe out your sole survivor?

[Rubik87]

Great job, Onco!!!

Wow, this is even worse(better) than my hypotesis. Not only bombs are not an equalizer, they are a dis-equalizer; you wish to use them when you have an advantage to improve it further more.

Considering also that they don't require a good torus position to be thrown from(they are global powers)..

Bombs and Smart Bombs are voted down for me.

[OncoByte]

Some other ways to look at the data:

Question: How many enemies can you expect to take out with a standard 16-bomb volley in an 8x10 arena?

Answer: This varies with the size of your opponent's squadron. The mean is shown below - half of the time you will hit more, half of the time you will hit less. The error bars are standard deviations - meaning that approximately 2/3 of the time, you the number of tori you hit will fall in this range. Note that Smart Bombs do not add that much even if you have a full squadron of 20 tori as is assumed in this example.


Question: What is the chance that you escape a 16-bomb volley unscathed?

Answer: Again this varies with the size of your squad. For smaller squadrons, the chance of surviving is pretty good. If you have 5 tori, for example, you will make it unscathed 35% of the time (78% of the time you lose less than 2 tori). With four tori left, you have a 56% chance of losing 1 or more tori (41% is the chance you lose exactly 1), but a 44% chance of remaining untouched.

[Handi]

Very nice graphs Onco, like usual (what program did you use?).

HOWEVER, don't make conclusions too quickly from statistics.
If I have 15 pieces and you 10, I will force you in exchanges that will cost lets say 5 pieces to each one of us. At 10 pieces left, I won't be able to pressure you as well anymore. Still, with 5 pieces versus 10, you are screwed.

Use bombs, blow 3 pieces from your enemy vs 1 of yours if you are a bit lucky. Result, I have 12 pieces left, you 9. Landscape is screwed so it's even harder for me to pressure you. Now you have much better chances for a comeback.

Also, If there are orbs in my territory and I have some well powered pieces, while you have practically nothing, it would be quite stupid not to use bombs.

You can hate smart bombs, but hating bombs to the point of never using them is... well, you know.

[OncoByte]

... I have an odd situation that I'd like to see...

16 Bomb volley, 8x10 board, 20 vs. 1, and obviously Smart Bombs probabilities need not be applied.

What are the chances that you obliterate a significant amount of opponent pieces (Not sure of a good # for this....a quarter? Half?) compared to the likelihood that you wipe out your sole survivor?

The chance that regular bombs nail your last piece is 18.2%.

If your opponent has a full squad of 20 tori - you will take out 3.64 of these on average (52% of the time you will take out 4 or more of his pieces). However, it is unlikely that you will take out much more than 4 as the odds for hitting more than that fall off rather fast.

The left column is the number of enemy tori you want to hit. The second column is the chance you hit exactly that number. The third column is the chance you hit that many or more.

Code: Select all

                 Cumulative
# Hit   Prob       Prob
   0    1.0%      100%
   1    6.1%       99.0%
   2   16.1%       92.9%
   3   24.6%       76.9%
   4   24.4%       52.2%
   5   16.5%       27.8%
   6   7.9%        11.3%
   7   2.7%         3.5%
   8   0.66%        0.79%
   9   0.115%       0.131%
  10   0.0144%      0.0158%
  11   0.00126%     0.00133%
  12   0.000074%    0.000077%
  13   0.0000028%   0.0000029%
  14   0.0000001%   0.0000001%
  15   0.0000000%   0.0000000%
  16   0.0000000%   0.0000000%


Here it is in graphical form.


In answer to your question - with a sole surviving torus, you have a roughly 50% chance of firing 3-4 volleys of Bombs without getting hit. On average, this will cull 9-11 of your opponent's 20 tori. Not too bad - but there's a 50% chance of instant Game Over.

[OncoByte]

Very nice graphs Onco, like usual (what program did you use?).
...
You can hate smart bombs, but hating bombs to the point of never using them is... well, you know.

I used Excel 2007 for both the VBA code and the graphs.

I do not hate Bombs - in fact, I love Bombs. They are a well balanced power even if they favor the player with more tori. When used well, they can remove unclaimed orbs and take away the high-ground advantage from an opponent. It's the Smart Bombs I am not a fan of.

[Rubik87]

Question: How many enemies can you expect to take out with a standard 16-bomb volley in an 8x10 arena?

Answer: This varies with the size of your opponent's squadron. The mean is shown below - half of the time you will hit more, half of the time you will hit less. The error bars are standard deviations - meaning that approximately 2/3 of the time, you the number of tori you hit will fall in this range. Note that Smart Bombs do not add that much even if you have a full squadron of 20 tori as is assumed in this example.

Show this expectation as a % of army instead of a #of tori please.

Once again, great job!

[Rydash]

This post could qualify as a Fan QReation, what with the charts.

Anyway, thanks for answering my question. However, it now becomes the more eternal "should you risk it?" question...Hmm.

[OncoByte]

Question: How many enemies can you expect to take out with a standard 16-bomb volley in an 8x10 arena?

Answer: This varies with the size of your opponent's squadron. The mean is shown below - half of the time you will hit more, half of the time you will hit less. The error bars are standard deviations - meaning that approximately 2/3 of the time, you the number of tori you hit will fall in this range. Note that Smart Bombs do not add that much even if you have a full squadron of 20 tori as is assumed in this example.
<GRAPH>

Show this expectation as a % of army instead of a #of tori please. Once again, great job!

I was very surprised by this finding! In retrospect, I should not have been.

The fraction of your opponent's squad you can expect to hit with a volley of Bombs is 18.2% no matter the size of the squad!

How the hell can this be if you are more likely to take out 25% or more of your opponent's squad when he has fewer tori?
Well, it turns out that you are also more likely to take out 10% or less of your opponents' squad when he has fewer pieces. The average fraction of the squad you hit remains the same.

Here it is shown graphically. Each curve is for an enemy squad of a given size. The fraction of that squad destroyed by bombs is shown on the x-axis. The chance that you destroy that fraction is shown on the y-axis.

As you can see, the more numerous the enemy squad, the more likely you are to hit the expected fraction of that squad. As the number of tori your opponent has goes down, the broader the probability curves becomes. This means that you are more likely to deviate from the expected value and take out either a larger or smaller fraction of his squad.

So Sanzo's original reasoning was not off base - just limited in scope. You can better predict how much damage you will suffer if you have a large number of pieces. That variability grows with smaller squadrons - making it both more likely you will suffer a large hit and more likely you will suffer a small hit. Take an extreme example - you have 3 tori left. You can either lose 0, 1, 2 or 3 tori. You are most likely to suffer no losses (54%), but your chance of losing a third of your squad is still 37%. Compare this to when you have 18 tori. The chance of your losing a third of your squad is 5%, but the chance of losing no tori is a mere 1.7%.

Finally - take the "Statistical Mean" with a grain of salt. The number of tori hit with a volley of bombs is an integer number, whereas the mean is not. The mean is useful when you consider the result across many volleys of Bombs. The discrete percentages are more useful when considering what to do in a given situation.

[Rubik87]

Very, very nice result!

So, each torus has 18.2% of being hit. Including your best tori (beneficiary, teach, learn, GQ, invisi+move diag etc.) wich are of course much more important than tori with just lower tile. Do you want to risk it? Basically yes, only if your opponent has much better powers(and if he knows how to use them ).

..I have some well powered pieces, while you have practically nothing, it would be quite stupid not to use bombs..

That's it.