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Subject: Math-Wing: Hit Probabilities rss

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Chris K.
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So, have you ever wondered, wether it's a better idea to take the shot at the Tie at Range 2 who only needs one more hit to be destroyed but has an evade token or rather the fresh one at Range 1 that you have a target lock on?

Well, wonder no more. Thanks to the magic of mathematics you can now find out.

Obviously there are piles of other considerations, but generally the chance to hit or to destroy your target plays a big role in deciding your shots. I was curious about how a few things interact with one another and I believe my results are pretty decent.

The probabilities are calculated for up to four dice attacking or defending and for getting at least 1, 2 or 3 hits.

Overall Conclusion:
The math works as it is seemingly intended.
An Evade Token is a better defense than a focus token.
Target Lock and Focus do exactly the same for your odds, except that Target Lock has a slightly higher chance of giving you a crit (the latter not shown in the tables).

Without further ado .. here's the tables:




Edit: The PDF Version is now approved: http://boardgamegeek.com/filepage/86555/math-wing-probabilit...
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Jay Little
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Christoph, you are a man after my own heart!
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Chris K.
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ynnen wrote:
Christoph, you are a man after my own heart!


Thank you so very much. A PDF Version is also submitted to the files section. It has all the table in those pictures with a slightly nicer colourcoding.
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Jeff Dunford
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chrisdk wrote:

An Evade Token is a better defense than a focus token.

... unless you're rolling lots of dice on defense and so is the attacker.
*edit* P.S. It isn't too uncommon to roll 5 defense dice (e.g. Imperial ship at Range 3 through an obstacle). With that many dice against 3 (or more) attack dice, you'd rather have a Focus than an Evade (although against 1 attack die you should always take the Evade - ignoring the offensive potential of the Focus token, of course).

chrisdk wrote:
Edit: Does anyone happen to know how I can make the images appear larger in the post?


Yes. Append "medium", "large" or "original" (without the quotes, no spaces) after the image id inside the square brackets. For example:
[ ImageID=1537549large ] (but remove the spaces)
 
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Jeff Dunford
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chrisdk wrote:
ynnen wrote:
Christoph, you are a man after my own heart!


Thank you so very much. A PDF Version is also submitted to the files section. It has all the table in those pictures with a slightly nicer colourcoding.


Any chance you can provide the spreadsheet? I think I see an error (2+ hits, Focus + Target Lock, 4:4 row) but it would be easier to check your calculation than do the calculation myself.
 
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Chris K.
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iNano78 wrote:
chrisdk wrote:
ynnen wrote:
Christoph, you are a man after my own heart!


Thank you so very much. A PDF Version is also submitted to the files section. It has all the table in those pictures with a slightly nicer colourcoding.


Any chance you can provide the spreadsheet? I think I see an error (2+ hits, Focus + Target Lock, 4:4 row) but it would be easier to check your calculation than do the calculation myself.


If you mean the 40,3% appearing twice, that is just because evade is only so slightly better that it dissappears in rounding in this case.

I'll see have to see if I find the time to clean the excel up well enough so that it makes sense for someone other than me. ;-)

Edit says: thanks for the tip to fix the images.
 
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Robert M.
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I've been wondering for a while how to display this kind of information in a sensible format (e.g., not 64 pages' worth of distributions). Nice work!
 
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Jeff Dunford
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chrisdk wrote:
iNano78 wrote:
chrisdk wrote:
ynnen wrote:
Christoph, you are a man after my own heart!


Thank you so very much. A PDF Version is also submitted to the files section. It has all the table in those pictures with a slightly nicer colourcoding.


Any chance you can provide the spreadsheet? I think I see an error (2+ hits, Focus + Target Lock, 4:4 row) but it would be easier to check your calculation than do the calculation myself.


If you mean the 40,3% appearing twice, that is just because evade is only so slightly better that it dissappears in rounding in this case.

I guess it threw me off because, from that table, the trend appears that Evade is better facing 2 attack dice (because then it's impossible to take 2+ hits), Focus is better if facing 3 attack dice (because the expected number of eyeballs is greater than 1), but they're approximately even facing 4 attack dice (despite the expected number of eyeballs).

I guess the trend would be more consistent (and more intuitive) in a table showing the expected number of damage cards taken, rather than the probability of taking a certain (whole) number of damage cards.
 
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Andrew H
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That's great stuff.

One table I would like to see is chances to hit with the different missiles...

e.g. the Proton Torps can chance one "eye" to a crit, so it's not quite the same odds as "Focus" but it's better than nothing...

Possible?
 
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Jeff Dunford
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I just finished a table of expectation values. It shows that Evade is better than Focus on defense for less than 4 defense dice, they're even for 4 defense dice, and Focus is better for 5+ defense dice. I'll upload it so people can compare it to the probability tables.

*edit* Somebody already beat me to it, although his analysis goes up to 5 attack dice and 4 defense dice.
 
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Chris K.
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iNano78 wrote:
chrisdk wrote:
iNano78 wrote:
chrisdk wrote:
ynnen wrote:
Christoph, you are a man after my own heart!


Thank you so very much. A PDF Version is also submitted to the files section. It has all the table in those pictures with a slightly nicer colourcoding.


Any chance you can provide the spreadsheet? I think I see an error (2+ hits, Focus + Target Lock, 4:4 row) but it would be easier to check your calculation than do the calculation myself.


If you mean the 40,3% appearing twice, that is just because evade is only so slightly better that it dissappears in rounding in this case.

I guess it threw me off because, from that table, the trend appears that Evade is better facing 2 attack dice (because then it's impossible to take 2+ hits), Focus is better if facing 3 attack dice (because the expected number of eyeballs is greater than 1), but they're approximately even facing 4 attack dice (despite the expected number of eyeballs).

I guess the trend would be more consistent (and more intuitive) in a table showing the expected number of damage cards taken, rather than the probability of taking a certain (whole) number of damage cards.


Hey, funnily enough.

You were right. The 3:4 Evade slot was off. It used the probabiliy for 3 defense dice rather than 4. At 12,8 to 12,6 they are nearly even again now.
 
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Dane Hitchins
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Some of your slots say 100%, how can you ever have a 100% probability to hit something? Even if you roll a million dice, they could come up blank. Rolling 4 dice against 1 doesn't seem like you should have a 100% chance of hitting the target. Am I wrong?
 
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Chris K.
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DaneHitchins wrote:
Some of your slots say 100%, how can you ever have a 100% probability to hit something? Even if you roll a million dice, they could come up blank. Rolling 4 dice against 1 doesn't seem like you should have a 100% chance of hitting the target. Am I wrong?


All values are rounded, so everything above 99.95% shows as 100.0%
 
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Dane Hitchins
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chrisdk wrote:
DaneHitchins wrote:
Some of your slots say 100%, how can you ever have a 100% probability to hit something? Even if you roll a million dice, they could come up blank. Rolling 4 dice against 1 doesn't seem like you should have a 100% chance of hitting the target. Am I wrong?


All values are rounded, so everything above 99.95% shows as 100.0%



But the value is really that high? 99.95% to get a hit?

I'm a little confused on what that means. Does it mean you have 99.95% chance to get one hit symbol that might be canceled by the opponent's dice, or does it means you have 99.95% chance to actually put a damage card on the enemy?
 
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Robert M.
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DaneHitchins wrote:
chrisdk wrote:
DaneHitchins wrote:
Some of your slots say 100%, how can you ever have a 100% probability to hit something? Even if you roll a million dice, they could come up blank. Rolling 4 dice against 1 doesn't seem like you should have a 100% chance of hitting the target. Am I wrong?


All values are rounded, so everything above 99.95% shows as 100.0%



But the value is really that high? 99.95% to get a hit?

I'm a little confused on what that means. Does it mean you have 99.95% chance to get one hit symbol that might be canceled by the opponent's dice, or does it means you have 99.95% chance to actually put a damage card on the enemy?

Looking at (for instance) the lowest cell on the left in the first table: when a ship with 4 Attack dice plus Target Lock and Focus targets a ship with 1 Agility die and no defense upgrades, the value (about 100%) represents the likelihood of actually generating at least one (1+) damage card.

udat wrote:
That's great stuff.

One table I would like to see is chances to hit with the different missiles...

e.g. the Proton Torps can chance one "eye" to a crit, so it's not quite the same odds as "Focus" but it's better than nothing...

Possible?

I've got that data, although it will take a while to format it--and I probably won't do as nice a job as the OP. (c:

I'll get it up ASAP, though.
 
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Andreas Krüger
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iNano78 wrote:
I just finished a table of expectation values. It shows that Evade is better than Focus on defense for less than 4 defense dice, they're even for 4 defense dice, and Focus is better for 5+ defense dice. I'll upload it so people can compare it to the probability tables.

*edit* Somebody already beat me to it, although his analysis goes up to 5 attack dice and 4 defense dice.


Don't forget that evade gives you a guaranteed success. Expectation values may be slightly skewed due to rolls with more focus than you need. Three eyes is not better than one eye if you only need one. So, you need also a high number of attack dice to make focus more valuable.
 
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tom brown
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they should really change the 100% to 99% as 100% isnt accurate in a dice game, nothing is ever 100%
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Jeff Dunford
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Thamos von Nostria wrote:
iNano78 wrote:
I just finished a table of expectation values. It shows that Evade is better than Focus on defense for less than 4 defense dice, they're even for 4 defense dice, and Focus is better for 5+ defense dice. I'll upload it so people can compare it to the probability tables.

*edit* Somebody already beat me to it, although his analysis goes up to 5 attack dice and 4 defense dice.


Don't forget that evade gives you a guaranteed success. Expectation values may be slightly skewed due to rolls with more focus than you need. Three eyes is not better than one eye if you only need one. So, you need also a high number of attack dice to make focus more valuable.


Expectation values are negative if you roll more evades (or modify more focuses into evades) than you need... but I got my spreadsheet to treat negatives as "0" (note that the link I posted preserves the negative values for use in his cumulative normal distribution curves). As for "skewing", the issue is at what point is the expectation value of eyes >= 1, and since there are 2 of them on a die, that number of defense dice is 4. If you're interested in variance, then an evade is a guaranteed success, while a focus is not guaranteed but occasionally allows you to cancel several successful attack dice. Also, all of this ignores that a Focus on a low-skill (Academy) pilot that doesn't get used for defense (assuming your opponent's ships are of higher skill) can be used for offensive purposes, giving it more flexibility - which, in my experience, is often worth it.

The key take-home message is that if you have a Agility 3 ship at Range 3 and behind an asteroid, you should take Focus rather than Evade.

ironman99 wrote:
they should really change the 100% to 99% as 100% isnt accurate in a dice game, nothing is ever 100%


Really? I think I'd rather see correct stats than incorrect stats. Perhaps the numbers should show more decimal places (and/or be switched to scientific/engineering notation). By the way, there are cases giving a 100% chance to hit in this game. For instance, a Concussion Missile with Focus (e.g. Vader's specialty) against a Y-wing with a critical that decreases its Agility value is guaranteed to score at least one hit (even if you roll all blanks).
 
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Robert M.
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udat wrote:
That's great stuff.

One table I would like to see is chances to hit with the different missiles...


"Hits", here, is defined as "hits + crits" to avoid complexity in presenting the data.



Three notes/caveats:

1) I didn't work out Homing Missiles or Assault Missiles because they work just like a regular attack with 4 dice. Ditto Cluster Missiles, with 3 dice. So the OP has already done the hard work there.

2) This doesn't include the impact of additional actions, like Focus or Target Lock. Since filling in some holes and formatting this is already an afternoon's worth of work, that may appear later or may not!

3) The table is composed of data collected from 1,000,000 simulated attacks for each case. This means the percentages and expected values may not be exact, although in the case of the percentages they're accurate to within +/- 0.1%.

EDIT: Forgot one.

4) For the percentage tables, the colors are assigned by quartile: green is for 75-100%, yellow is for 50%-75%, orange is for 25%-50%, and red (fuschia?) is for 0-25%.
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Chris K.
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Focus vs Evade:

I belief that is also why it is designed this way: Evade can only be used for defense and is therefor better at it than the more generalised focus that can be used for something else as well. Just like a target lock is slightly better than a focus by giving you a slightly better chance to generate criticals and being sustainable if it is unused this round.

99% vs 100%
We are talking about taking probabilities as a decision guide in a game. Rounding to 10s of Percents would be accurate enough to do that because you will never roll often enough for more exact numbers to matter.

Personally, if there isn't a difference of at least 20% between two options I consider them "near identical" anyway and look for other factors to make my decision on.

Going with exactness to the 3rd or 4th digit is why a lot of people lable "Math-Winging" obnoxiously anal, since "no matter the chance, the dice will still screw you".

And yes, in the best Spirit of Discworld, One-in-a-Million-Chances come up nine times out of ten, so I am fully aware that everyone has stories to tell about times when they were attacking 4 vs 1 and didn't put a damage card on the target which means that all statistics math is totally worthless ;-)

As to wether that 99,95%+ is accurate:
Target Lock and Focus means that every single die has a chance of over 90% to roll a hit and you roll 4 of them. It is pretty improbable to get only one hit out of that.
Your target has only one die. So at best can cancel one hit, and even that only in slightly more than 60% of the rolls.

So yes, 99,95+% of putting at least one card on him is accurate.
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Dan Q
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Dumb question: Is it legal to bring and reference these tables at a tournament?
 
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Chris K.
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Supagoat wrote:
Dumb question: Is it legal to bring and reference these tables at a tournament?


Personally I'd consider it poor sportmanship and dragging out the game unnecessarily. Not sure about any rules against it, though.

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Robert M.
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chrisdk wrote:
Supagoat wrote:
Dumb question: Is it legal to bring and reference these tables at a tournament?


Personally I'd consider it poor sportmanship and dragging out the game unnecessarily. Not sure about any rules against it, though.

I'm with Christoph: it's probably bad manners, although I don't think it's explicitly forbidden. You might instead use the table to formulate some simple guidelines or targeting priorities you can remember easily.
 
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Andreas Krüger
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Tables: For a noob like me probably better than turning the own maneuver wheel a few times in order to check which available maneuvers are green.... Good players will know the table anyway.

Oops, confused this thread with another subscription, so my comment does not make any sense here :-).
 
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Andreas Krüger
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I wonder how much effect these crits really have. Most statistical analyses neglect them, and this may result in underestimating the power of rebel shields and upgrades like proton torpedoes and marksmanship. Since a critical is the last die that is canceled by evades, it is not that unlikely.
 
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