Re: Statistical odds of AA guns + standard deviations associated low/high sample

[calvinhobbesliker]

Can we stop arguing about the probabilities? It’s getting repetetive and isn’t going anywhere

[Imperious Leader]

How on earth can sub hitting with 2 an a des hitting with 2 have fighting against each other have different odds? Roflmao, that is just so wrong on first sight.
The 40% is correct, you need to set your Simulator on throwing more often, and you’ll see, you’ll get closer to 40%. (A simulator doesn’t do math, it just rolls dice, so it’ll be of a bit of the true odds, unless you do a huge number of rolls (like a million))

I posted the wrong numbers.

anyway. The result is like 39.7 to 41.2% with 10,000 events

http://www.dskelly.com/misc/aa/aasim.html

But you see the statistical variation also shows the differences depending on throws.

with 1,000 runs the odds are 40.8 for the sub
with 5,000 runs the odds are 39.9 for the sub
at 10,000 it goes to 39.7

This is why with only 7 rolls i don’t see why you can make a claim of 67%+ Not enough numbers.

http://en.wikipedia.org/wiki/Standard_deviation

Only with large numbers can the truth of plane loses be qualified.

here is another result: This accounts for 8 different rolls and the odds of killing 0-8 planes

To kill one plane it shows: 18.98% which is much closer to my 16.6% than your 67%

Probability % # units / losses
  23.12% 8: 4 Fig, 4 Bom. no units. : 0 IPCs
  18.98% 7: 3 Fig, 4 Bom. 1 Fig. : 10 IPCs
  18.27% 7: 4 Fig, 3 Bom. 1 Bom. : 15 IPCs
  14.52% 6: 3 Fig, 3 Bom. 1 Fig, 1 Bom. : 25 IPCs
  5.48% 6: 2 Fig, 4 Bom. 2 Fig. : 20 IPCs
  5.42% 6: 4 Fig, 2 Bom. 2 Bom. : 30 IPCs
  4.87% 5: 3 Fig, 2 Bom. 1 Fig, 2 Bom. : 40 IPCs
  4.67% 5: 2 Fig, 3 Bom. 2 Fig, 1 Bom. : 35 IPCs
  0.68% 5: 4 Fig, 1 Bom. 3 Bom. : 45 IPCs
  0.8% 5: 1 Fig, 4 Bom. 3 Fig. : 30 IPCs
  1.33% 4: 2 Fig, 2 Bom. 2 Fig, 2 Bom. : 50 IPCs
  0.65% 4: 3 Fig, 1 Bom. 1 Fig, 3 Bom. : 55 IPCs
  0.56% 4: 1 Fig, 3 Bom. 3 Fig, 1 Bom. : 45 IPCs
  0.06% 4: 4 Fig. 4 Bom. : 60 IPCs
  0.01% 4: 4 Bom. 4 Fig. : 40 IPCs
  0.24% 3: 2 Fig, 1 Bom. 2 Fig, 3 Bom. : 65 IPCs
  0.23% 3: 1 Fig, 2 Bom. 3 Fig, 2 Bom. : 60 IPCs
  0.04% 3: 3 Fig. 1 Fig, 4 Bom. : 70 IPCs
  0.04% 3: 3 Bom. 4 Fig, 1 Bom. : 55 IPCs
  0.01% 2: 2 Bom. 4 Fig, 2 Bom. : 70 IPCs
  0.01% 2: 1 Fig, 1 Bom. 3 Fig, 3 Bom. : 75 IPCs
  0.01% 1: 1 Fig. 3 Fig, 4 Bom. : 90 IPCs

http://frood.net/aacalc/?mustland=0&abortratio=0&saveunits=0&strafeunits=0&AA=on&aInf=&aArt=&aArm=&aFig=4&aBom=4&aTra=&aSub=&aDes=&aCar=&aBat=&adBat=&dInf=&dArt=&dArm=&dFig=&dBom=&dTra=&dSub=&dDes=&dCar=&dBat=&ddBat=&ool_att=Bat-Inf-Art-Arm-Tra-Sub-SSub-Fig-JFig-Des-Bom-HBom-Car-dBat&ool_def=Bat-Inf-Art-Arm-Tra-Sub-SSub-Bom-HBom-Des-Fig-JFig-Car-dBat&battle=Run&rounds=&reps=10000&luck=pure&ruleset=Revised&gameid=&password=&turnid=&territory=&round=1&pbem=

[kcdzim]

@Imperious:

To kill one plane it shows: 18.98% which is much closer to my 16.6% than your 67%

Um, no, and IL, despite a decent tactical mind and some very nice Illustrator skills, you’re killing me here.  For the love of god, you need to take a statistics class (or refresh your statistics notes).

This has never been a calculation of losing ONE plane, and no one has ever argued that that is anything BUT 1 in 6, as that’s the freaking dice roll!   A one in six chance of the pip coming up is ALWAYS a 1 in 6 chance!  This is a combined odds calculation of the chances of losing ANY plane.  The calculation you just posted confirms all this, as muddied as it is.  Why on earth would you add in two different types of planes?   That just makes the whole thing even more complicated because now we have completely unecessary data with three possible types of hits (on bombers or on fighters or on both).   If you had bothered to look at the other instances you posted instead of just copying and pasting (odds of losing 2, odds of losing 3, odds of losing 4) you would notice that it freaking CONFIRMS the 67%.  By simply looking at the odds of losing 0 fighters and 0 bombers, you inadvertantly prove what we’re saying.  In 8 planes, you have a 76.88% chance of losing SOMETHING, if you have a 23.12% chance of losing NOTHING (because everything adds to 100%).

The general formula for rolling at least one hit in n rolls is 1 - (5/6)^n.  Yes, you are correct, that each roll is 1/6 (16%) and yes, you are correct that each roll has no impact on the previous or the next.  But bringing standard deviation and statistical variation for number of throws is completely unnecessary, and apparently is confusing you because you appear to think it’s relevant.

In the infinite number of instances where 6 planes are fired at by AA, in 67% of them at least one plane will die. It might be an instance of one plane, but it could be an instance of all 6 planes (which is still at least one plane).  It’s the COMBINED odds.  Not just the odds of losing ONE plane.  and 33% of the time (a whole 1 in 3) you lose NOTHING (which apparently you remember more often than the times where you lose more than one plane - good on you, it’s fun to revel in our enemies’ destruction).

When AA odds calculators do there stuff, and we end up with a calculation that has a win percentage, and that win percentage includes all possible results that equal a win.  They do not give you the odds of only one type of win, or only one type of result.  Whether it’s the odds of taking the win with only one loss, two losses, etc etc, it’s all in the combined odds.  Stop getting hung up on your dangerously incomplete understanding of statistics.

By the by, in Vegas, if you WERE to bet on the same number, eventually you WOULD win.  The reason Vegas works is despite the statistical inevitability that eventually, with an infinite number of rolls you will eventually result a win (in fact, an infinite number of wins), no one has infinite money, and the odds are better that you’ll lose more than you can afford to before you get an UNLIKELY win early.

[jim010]

You people are giving me a headache …

I’m going to bed.

[Lydian]

@Imperious:

How on earth can sub hitting with 2 an a des hitting with 2 have fighting against each other have different odds? Roflmao, that is just so wrong on first sight.
The 40% is correct, you need to set your Simulator on throwing more often, and you’ll see, you’ll get closer to 40%. (A simulator doesn’t do math, it just rolls dice, so it’ll be of a bit of the true odds, unless you do a huge number of rolls (like a million))

I posted the wrong numbers.

anyway. The result is like 39.7 to 41.2% with 10,000 events

http://www.dskelly.com/misc/aa/aasim.html

But you see the statistical variation also shows the differences depending on throws.

with 1,000 runs the odds are 40.8 for the sub
with 5,000 runs the odds are 39.9 for the sub
at 10,000 it goes to 39.7

This is why with only 7 rolls i don’t see why you can make a claim of 67%+ Not enough numbers.

http://en.wikipedia.org/wiki/Standard_deviation

Only with large numbers can the truth of plane loses be qualified.

Still getting it wrong.
When I say the outcome of 7 AA rolls is 72% at least one hit, then I’m not making that assumption by rolling qa dice seven times, that assumption is made on rolling 7 dice infinite times.
A coin flip is 50%, 7 rolls at least one 1 is 72%, that is the probability of that event.

If I roll 7 AA rolls infinitely, then it’ll show 72% of the rolls containing at least one hit.
If I only roll those 7 dice once, ten or a hundred times, that is where the standard derivation comes in, since then, not neccassarily 72% of the 7 rolls will contain at least one 1, but the number would still be around that 72%, most likely at least.
7 dice rolls is not where I deduct the 72%, hell I don’t roll any dice, I calculate the odds. And when I calculate them, I get the result which throwing the dice an infinite number would yield me.
And that is the only % to take when planning, since standard derivation will obscure these 72% in my games, but it, or in that area, still is the likliest event to come up in my AA games.
The number 7 AA dice is not equal to the number of times I throw these 7 dice, that’s an important difference.
I can calculate the odds of tossing a coin once (50% tails), that doesn’t mean one toss is where I calculate the odds from.
When you calculate the odds, you get the correct %, and if you like (but it’s uneccessary in A&A), you can also calculate the standard derivation.
You can also calculate the odds of a specific situation coming up (e.g. exactly two hits, or less than 3 hits, if you can afford losing two planes for example). This may be neccassary at times, but with our AA example we only calculated the odds of at least one AA hit, which is, as many people have posted the formula, really easy.

Using your sim shows how you won’t get the exact odds, since with what you let them roll it is not enough. The true odds are 40/40/20, and logic dictates, that des sub and des must have the same chance of winning, since they have equal stats.
Still, the Simulator doesn’t give any odds, but something that is near the truth. And that’s usually good enough for AA games.

@Imperious:

here is another result: This accounts for 8 different rolls and the odds of killing 0-8 planes

To kill one plane it shows: 18.98% which is much closer to my 16.6% than your 67%

Probability % # units / losses
  23.12% 8: 4 Fig, 4 Bom. no units. : 0 IPCs
  18.98% 7: 3 Fig, 4 Bom. 1 Fig. : 10 IPCs
  18.27% 7: 4 Fig, 3 Bom. 1 Bom. : 15 IPCs
  14.52% 6: 3 Fig, 3 Bom. 1 Fig, 1 Bom. : 25 IPCs
  5.48% 6: 2 Fig, 4 Bom. 2 Fig. : 20 IPCs
  5.42% 6: 4 Fig, 2 Bom. 2 Bom. : 30 IPCs
  4.87% 5: 3 Fig, 2 Bom. 1 Fig, 2 Bom. : 40 IPCs
  4.67% 5: 2 Fig, 3 Bom. 2 Fig, 1 Bom. : 35 IPCs
  0.68% 5: 4 Fig, 1 Bom. 3 Bom. : 45 IPCs
  0.8% 5: 1 Fig, 4 Bom. 3 Fig. : 30 IPCs
  1.33% 4: 2 Fig, 2 Bom. 2 Fig, 2 Bom. : 50 IPCs
  0.65% 4: 3 Fig, 1 Bom. 1 Fig, 3 Bom. : 55 IPCs
  0.56% 4: 1 Fig, 3 Bom. 3 Fig, 1 Bom. : 45 IPCs
  0.06% 4: 4 Fig. 4 Bom. : 60 IPCs
  0.01% 4: 4 Bom. 4 Fig. : 40 IPCs
  0.24% 3: 2 Fig, 1 Bom. 2 Fig, 3 Bom. : 65 IPCs
  0.23% 3: 1 Fig, 2 Bom. 3 Fig, 2 Bom. : 60 IPCs
  0.04% 3: 3 Fig. 1 Fig, 4 Bom. : 70 IPCs
  0.04% 3: 3 Bom. 4 Fig, 1 Bom. : 55 IPCs
  0.01% 2: 2 Bom. 4 Fig, 2 Bom. : 70 IPCs
  0.01% 2: 1 Fig, 1 Bom. 3 Fig, 3 Bom. : 75 IPCs
  0.01% 1: 1 Fig. 3 Fig, 4 Bom. : 90 IPCs

http://frood.net/aacalc/?mustland=0&abortratio=0&saveunits=0&strafeunits=0&AA=on&aInf=&aArt=&aArm=&aFig=4&aBom=4&aTra=&aSub=&aDes=&aCar=&aBat=&adBat=&dInf=&dArt=&dArm=&dFig=&dBom=&dTra=&dSub=&dDes=&dCar=&dBat=&ddBat=&ool_att=Bat-Inf-Art-Arm-Tra-Sub-SSub-Fig-JFig-Des-Bom-HBom-Car-dBat&ool_def=Bat-Inf-Art-Arm-Tra-Sub-SSub-Bom-HBom-Des-Fig-JFig-Car-dBat&battle=Run&rounds=&reps=10000&luck=pure&ruleset=Revised&gameid=&password=&turnid=&territory=&round=1&pbem=

How can someone be so blind, that just was an classical own goal, but thank you for posting that.
Your table shows the odds of no plane being hit at 23,12%. That means the chance of losing at least one plane (=rolling at least one 1, what we all have been talking about) is 76,12%.
If you calculate the odds, you get 76,74% for at least one 1 throwing 8 dice. And that are the true odds.
Again, the Simulator is a little off, but the result is still very much near the truth.

[HolKann]

I didn’t read all of the probabilities discussion, but I think I got the crux of the problem (with Lydian thinking pointing out math flaws will convince IL, and with IL not clearly stating what the point they disagree upon).

The funny thing is, both sides of the discussion are right. You both agree that one AA shot has a ~17% chance of hitting, and that ON AVERAGE = USING THE LAW OF LARGE NUMBERS 6 AA shots yield ~67% chance of hitting at least one plane. This is basic maths, and both IL and Lydian get this, no use arguing about this any longer.

What IL is trying to say, is that in 1 AA game, the law of very large numbers doesn’t work. And this is what Lydian disagrees about. This results in sentences like “what do you think is the probability of hitting at least 1 plane with 6 AA shots?”. Lydian defines “probability” the mathematical way, “on average”, “the chance the event happening when repeated and infinite amount of times”. IL refuses to define probability this way, arguing that in one random test, the a particular “on average” result is not very likely (which is mathematically correct!), so he says the only thing you are sure about during a small number of rolls, is that 1 AA shot hits in 17% of cases. Both views are correct, but look at things differently (use different definitions).

What I like to point out, is that the mathematical probability is a valid tool to assess combat moves, and can be used to argue whether a strategy will work in the long run (or using no/low luck). This is why IL’s 42% isn’t enough to do the G2 Sealion. IL argues that he doesn’t have to, as his main goal is letting the Italian navy survive. Both are correct again.

Now:
@calvinhobbesliker:

Can we stop arguing about the probabilities? It’s getting repetetive and isn’t going anywhere

[Lydian]

@HolKann:

What I like to point out, is that the mathematical probability is a valid tool to assess combat moves, and can be used to argue whether a strategy will work in the long run (or using no/low luck). This is why IL’s 42% isn’t enough to do the G2 Sealion. IL argues that he doesn’t have to, as his main goal is letting the Italian navy survive. Both are correct again.

That is the whole point:
Of course in a game of A&A everything can happen, 1 inf can defeat 3 arms, but if my strategy relies on unlikely events it’s a bad strategy.
Since the 7 AA shots will take down a plane 72% of the time, combined with odds of at least on battle going wrong in G1, IL’s strategy isn’t good enough.
Especially since it doesn’t even save the Italian navy, as England can use the TAC to sink it and still have the odds on its side in the UK battle.

And, when planning your moves and battles, the odds which, either the mathematical formula or throwing a large number of dice (preferably done by a simulator), which come up with pretty much the same result (and the error being on the sim’s side, since he doesn’t throw enough dice), are the ones to be guided by.
And they say, that the proposed strategy won’t work even 20% of the time, and as such is no good.
Jim’s is much more interesting, and I’m curious whether he stays above 40%, or even gets over 50%. Sadly it doesn’t neccessarily save the Italian navy either.

[Imperious Leader]

What IL is trying to say, is that in 1 AA game, the law of very large numbers doesn’t work. And this is what Lydian disagrees about. This results in sentences like “what do you think is the probability of hitting at least 1 plane with 6 AA shots?”. Lydian defines “probability” the mathematical way, “on average”, “the chance the event happening when repeated and infinite amount of times”. IL refuses to define probability this way, arguing that in one random test, the a particular “on average” result is not very likely (which is mathematically correct!), so he says the only thing you are sure about during a small number of rolls, is that 1 AA shot hits in 17% of cases. Both views are correct, but look at things differently (use different definitions).

Yes exactly.

All i have been trying to say is the statistical variation is not reached in 7 rolls, the sample is not great enough to produce the quoted results.

If you notice that you can run the numbers for 1K, 5K 10K number of times and each time the value changes. This is because the large number sample gets you this answer.

I said many times it might be less or more than the stated 67% and anything in between. You can of course calculate the value, but its not correct in terms of an actual produced result.

Out of any number of test runs of rolling 8 dice and only 8 dice, you will not get to an exact %. Its too small of a sample. If you roll out 1 million dice say even once, you will get the 'scientifically clean results"

Otherwise, your stuck with many variations in results and that average is not “pure”. Again the sample is too small.

BTW my sealion concept is only for stopping the Italian fleet nuke. Its not a real invasion, but a threat of one hopefully enough to make them not kill Italy on UK1.

A real sealion plan is for this G3, but too me it makes it near impossible to sway UK from killing the Italians. This is because the AP builds are not enough to commit a G2 and that gives too much time for UK to get back to UK after killing Italy.

And yes after looking and retyping Jim’s concept, i can also come to that conclusion. Its a better plan. But that was not of issue.

[jim010]

I would like to point out that just going after Russia and defending France, there are far more battles involved over a number of turns.

The likelyhood of all those battles going your way is even less.

[Lydian]

@Imperious:

All i have been trying to say is the statistical variation is not reached in 7 rolls, the sample is not great enough to produce the quoted results.

If you notice that you can run the numbers for 1K, 5K 10K number of times and each time the value changes. This is because the large number sample gets you this answer.

I said many times it might be less or more than the stated 67% and anything in between. You can of course calculate the value, but its not correct in terms of an actual produced result.

Out of any number of test runs of rolling 8 dice and only 8 dice, you will not get to an exact %. Its too small of a sample. If you roll out 1 million dice say even once, you will get the 'scientifically clean results"

Nobody has ever doubted that throwing 8 dice once, or 100, or even 1000 times is too small a sample.
This may be the crux of misunderstanding. It was pointed out more than once, but again.
The 67% are not determined by rolling 6 dice once, but by calculating what the odds of at least one 1 would be if you rolled 6 dice infinitely.
And your example with 8 dice proves that the formula is correct (which is a universally accepted mathematical truth anyway).
The calculated odds are 76,74% , the sim comes up with 76,12%.

And I think it is clear to everyone, that with the few rolls in a game of A&A that the odds will not neccessarily show perfectly, but usually at least near them most of the time and, most important, it is the number which should guide your planning.

@Imperious:

BTW my sealion concept is only for stopping the Italian fleet nuke. Its not a real invasion, but a threat of one hopefully enough to make them not kill Italy on UK1.

A real sealion plan is for this G3, but too me it makes it near impossible to sway UK from killing the Italians. This is because the AP builds are not enough to commit a G2 and that gives too much time for UK to get back to UK after killing Italy.

And yes after looking and retyping Jim’s concept, i can also come to that conclusion. Its a better plan. But that was not of issue.

Agreed. But as a UK player I’d still sink the Italian fleet, since, when consulting the odds, UK is safe (not in a 100% meaning, but over 60% if G1 works out) without the TAC.

@Jim
Agreed. But in the battles against Russia it’s not that big of a deal whether you lose one inf or more, or don’t take a territory you should every other round. Until the final battle it’s usually only the stack moving forward and crushing a single inf or territory exchanges. The greater strategy works even with setbacks, whereas, as far as I understand, each battle to succed is pretty much vital for the invasion plan, so not even one can fail.

But nonetheless, good work. I also kept this odds discussion out of your thread, as to not infere with the discussion of that idea proper. I’m locking forward to see where it ends up.
Unfortunately, besides the fear of a Sealion G2, which seems hardly doable with favourable odds, there doesn’t seem to be a way of stoping the sinking of the Italian navy. Sadly…
Allthough, if the risk of G3 is great enough, it might keep the English player from putting his planes at risk against the Italian fleet.

[SgtBlitz]

@Imperious:

People: i already posted the solution. This stops any credible attack against the main German fleet. Transports protected.

Attack SZ 106 with 1 sub ( UK has 1 DD) 2 vs. 2. 40.6% to 39.5%

Attack SZ 109 1 fighter vs. 1 DD 3 vs. 2. 48.9% to 26% ( both die at 25.1%)

Attack SZ 110 2 subs, 1 tactical, 1 fighter, 1 bomber vs. 1 BB, 1 DD ( should win)
99.5%

Attack SZ 111 2 subs, 1 tactical, 1 Fighter, vs. 1 BB, 1 CA ( should win)
85.7%

Attack SZ 112 1 BB, 1 tactical, 1 Fighter vs 2 CA ( should win hit on BB)
89%

Notes:
The cruiser makes my BB and CV with 2 fighters protected against 3 of his fighters attacking on UK1

His navy is blocked entirely against SZ 112. I have 2 subs each on each side to block a DD from coming in.

I could fix SZ 109 attack and replace with bomber, and put the fighter with the BB attack. This cuts out the DD in SZ 109 from coming to the other side with 3 fighters against my main fleet.

The first attack IS a coin flip, but the others are not. I expect to lose 2-3 subs and a BB hit. If i roll down i expect to lose a fighter or bomber in SZ 109

One CA blocks at SZ 104 the subs block either DD.

If i lose both subs in one battle, i think i can still win against 3 fighters and DD against a 2 hit CV BB and 2 fighters. But its set up so that i should just lose one SS in each combat.

This both protects my main fleet that is coming out, causes UK to avoid attacking Italian fleet, Gives Germany a chance for Sealion, kills most of UK’s fleet, and kills the balance of UK’s fleet on the next turn if they choose to attack my CA on UK1.

Do the subs really block the destroyers?  Is there a special rule that prevents the destroyers from moving past sea zones with subs in them?  Thought they could be ignored in AA40.

You are only also going up against the UK with 3 transports.  So that means two tanks, 3 inf and an arty are on the boats G2.  ALL your airforce is required to survive to try to win the G2 Sealion.  That’s 7 aircraft + 6 land units against 11, possibly 13 land units and 3-4 planes defending.  Not a chance.

IF you build 8-9 transports on G2, you have a pretty good chance of taking London on G3.  But since this is even MORE unviable in Global40 than in Europe40 (with an 82 IPC USA incoming), I’d say Sealion is out of the equation unless the UK player just doesn’t know what he’s doing.

[kcdzim]

@SgtBlitz:

@Imperious:

I have 2 subs each on each side to block a DD from coming in.

Do the subs really block the destroyers?  Is there a special rule that prevents the destroyers from moving past sea zones with subs in them?  Thought they could be ignored in AA40.

Subs are NEVER able to block destroyers.  It’s the attackers prerogative, ALWAYS, whether to attack or ignore a sub.  Destroyers block subs, NOT vice versa.

I’m not sure if this an old post of IL’s that’s been quoted ad nauseum or he copy/pasted himself without a correction, but it’s been posted several times with this same oversight.  And it’s been corrected several times.

[Imperious Leader]

calculating what the odds of at least one 1 would be if you rolled 6 dice infinitely.

Yes thats what i was saying.

Subs are NEVER able to block destroyers.

Right. The thing is his DD’s are not in reach: I sunk the one with good odds off the western UK coast, and the Gibraltar fleet is blocked from attacking my SZ 112 fleet.

But the point is 4 out of 5 subs are close to my main fleet and under my fleets protection. They are not all over the place and cannot be picked off easily.

And Yes Sgt. is correct. I posted it in different threads because as you know the Sealion thread is in many places.

[ozimek1]

Please rename this thread to “Does endless statistics discussion break the ability to have a meaningfull discussion of Sealion”  :wink:

Seriously it’s been a nightmare to try to distill any strategic insight from this thread.

[Imperious Leader]

OK you want me to split the topic? I can do this.