Probability with an Abacus: Understanding SGLs

[Heroic]

This article provides a brief refresher course in probabilities that will help to understand the appearance of SGLs a little better. It will also illustrate noteworthy differences between the chances of an SGL for a scientific civ and a normal civ as the game progresses.

The Basics:
Most everyone knows that Scientific civs get a 5% chance of an SGL whereas other civs only get 3%. That means you have a 3% or 5% chance of him appearing whenever you research a technology that nobody else has. That is important to remember since it means that techs you find in huts will not help you get a SGL. In fact, huts can actually hurt your chances of getting an SGL if they give you a tech that nobody else has - because that leaves you with one less tech that gives a chance at a SGL. The same is true if your enemy finds a tech in a hut that nobody else has....especially if you happen to be researching it. In short, unique hut technology (no matter who finds it) is the enemy of SGL generation. So why do we all care about SGL's so much? Because the 1-turn wonder can be devastating at giving a crucial wonder at a crucial time...thus wasting an ememies shields spent in building, and also providing a unique advantage. An early (pyramids/ Zeus) and sometimes a late wonder (Lighthouse GreatWall) can often make a big difference.

The Myth:
So, how do we understand that 3 or 5 percent? Sadly, many of us forget and try and use a poor rule of thumb like just adding it to figure out our chances of an SGL. This logic goes something like this. "Hmm if i get 3 unique techs i'll have a 15% chance of a leader...and heck if i can get close to 20...I'll have him for sure". Unfortunately this is not quite true, but fortunately there is a way to know what your chances are! Some people also seem to be surprised because a SGL seems to show up in more than 5% (1 out of 20 games) but we shall see that this is no surprise at all.

The Truth:
Lets consider a simple example like "What is the probability of a SCI civ getting at least 1 SGL after 2 researches?"
We'll say that "L" = "we get a leader" and "N" = "we don't". In 2 turns, there are four possilbe outcomes:

Case 1 LL
Case 2 LN
Case 3 NL
Case 4 NN

We now substitute .95 i.e. 95% wherever we see N and .05 i.e. 5% wherever we see L. Then we multiply each line to see the probability of that case. As a check, you will note that the last column adds to 1 ..... meaning it accounts for 100% of the possibilities.

Case 1 .05 X .05 = .0025
Case 2 .05 X .95 = .0475
Case 3 .95 X .05 = .0475
Case 4 .95 X .95 = .9025

Case 1 tells us our chances of getting 2 SLGs in 2 turns is very small. 1/4 of a percent in fact! Case 2 tells us our chances of getting a SGL and then no SGL are 4.75% The chances are the same for getting no SGL and then getting an SGL. And we have a 90.25% chance of getting no SGL either time.

So back to our question.....what is the probability of getting at least 1 SGL in 2 unique techs? Well Lines 1-3 all give at least 1 SGL....so we add them, which gives us a 9.75% chance of getting an SGL. Note, 9.75% not 10% as we had supposed before.....this difference is even bigger if we had been using 3% instead of 5%. The difference will also get much bigger the more advances we consider.

Now, if you're like me, you dont wanna have to plunk out all the math unless you have Excel or something. Fortunately there is an easier way to figure out the probabilities. Look at line 4 ( the probability of no SGL in 2 turns). Our 9.75% = 100% - 90.25%. Cool so now we have a short cut at finding the probability of at least one SGL.

The process:
Step 1. F = 1 - SGL% which will give us either .95 or .97
Step 2. T = number of techs we research first
Step 3. N = F^T ( so, if t were 2 then N could be .95 squared)
Step 4. Chance of at least 1 SGL = 1 - N

[Heroic]

The Scientific Table:
Yes. I know nobody wants to do these by hand all the time. So here is a table for the probability of at least 1 SGL for a scientific civ. I have taken it out to only 50 because honestly how many of us research that many techs anyway?

#Techs ----- Prob (no SGL) ----- Prob (1+ SGL)
1 ----- .950000000000000 ----- 5.0%
2 ----- .902500000000000 ----- 9.8%
3 ----- .857375000000000 ----- 14.3%
4 ----- .814506250000000 ----- 18.5%
5 ----- .773780937500000 ----- 22.6%
6 ----- .735091890625000 ----- 26.5%
7 ----- .698337296093750 ----- 30.2%
8 ----- .663420431289062 ----- 33.7%
9 ----- .630249409724609 ----- 37.0%
10 ----- .598736939238379 ----- 40.1%
11 ----- .568800092276460 ----- 43.1%
12 ----- .540360087662637 ----- 46.0%
13 ----- .513342083279505 ----- 48.7%
14 ----- .487674979115530 ----- 51.2%
15 ----- .463291230159753 ----- 53.7%
16 ----- .440126668651766 ----- 56.0%
17 ----- .418120335219177 ----- 58.2%
18 ----- .397214318458218 ----- 60.3%
19 ----- .377353602535308 ----- 62.3%
20 ----- .358485922408542 ----- 64.2%
21 ----- .340561626288115 ----- 65.9%
22 ----- .323533544973709 ----- 67.6%
23 ----- .307356867725024 ----- 69.3%
24 ----- .291989024338773 ----- 70.8%
25 ----- .277389573121834 ----- 72.3%
26 ----- .263520094465742 ----- 73.6%
27 ----- .250344089742455 ----- 75.0%
28 ----- .237826885255332 ----- 76.2%
29 ----- .225935540992566 ----- 77.4%
30 ----- .214638763942937 ----- 78.5%
31 ----- .203906825745791 ----- 79.6%
32 ----- .193711484458501 ----- 80.6%
33 ----- .184025910235576 ----- 81.6%
34 ----- .174824614723797 ----- 82.5%
35 ----- .166083383987607 ----- 83.4%
36 ----- .157779214788227 ----- 84.2%
37 ----- .149890254048816 ----- 85.0%
38 ----- .142395741346375 ----- 85.8%
39 ----- .135275954279056 ----- 86.5%
40 ----- .128512156565103 ----- 87.1%
41 ----- .122086548736848 ----- 87.8%
42 ----- .115982221300006 ----- 88.4%
43 ----- .110183110235005 ----- 89.0%
44 ----- .104673954723255 ----- 89.5%
45 ----- .099440256987092 ----- 90.1%
46 ----- .094468244137738 ----- 90.6%
47 ----- .089744831930851 ----- 91.0%
48 ----- .085257590334308 ----- 91.5%
49 ----- .080994710817593 ----- 91.9%
50 ----- .076944975276713 ----- 92.3%

[Heroic]

The Primitive Table
This table is similar to the first except that it is the probability of 1+ SGL for non-Scientific civs.

#Techs ----- Prob(no SGL) ----- Prob(1+ SGL)
1 ----- 0.970000000000000 ----- 3.0%
2 ----- 0.940900000000000 ----- 5.9%
3 ----- 0.912673000000000 ----- 8.7%
4 ----- 0.885292810000000 ----- 11.5%
5 ----- 0.858734025700000 ----- 14.1%
6 ----- 0.832972004929000 ----- 16.7%
7 ----- 0.807982844781130 ----- 19.2%
8 ----- 0.783743359437696 ----- 21.6%
9 ----- 0.760231058654565 ----- 24.0%
10 ----- 0.737424126894928 ----- 26.3%
11 ----- 0.715301403088080 ----- 28.5%
12 ----- 0.693842360995438 ----- 30.6%
13 ----- 0.673027090165575 ----- 32.7%
14 ----- 0.652836277460607 ----- 34.7%
15 ----- 0.633251189136789 ----- 36.7%
16 ----- 0.614253653462686 ----- 38.6%
17 ----- 0.595826043858805 ----- 40.4%
18 ----- 0.577951262543041 ----- 42.2%
19 ----- 0.560612724666750 ----- 43.9%
20 ----- 0.543794342926747 ----- 45.6%
21 ----- 0.527480512638945 ----- 47.3%
22 ----- 0.511656097259776 ----- 48.8%
23 ----- 0.496306414341983 ----- 50.4%
24 ----- 0.481417221911724 ----- 51.9%
25 ----- 0.466974705254372 ----- 53.3%
26 ----- 0.452965464096741 ----- 54.7%
27 ----- 0.439376500173838 ----- 56.1%
28 ----- 0.426195205168623 ----- 57.4%
29 ----- 0.413409349013565 ----- 58.7%
30 ----- 0.401007068543158 ----- 59.9%
31 ----- 0.388976856486863 ----- 61.1%
32 ----- 0.377307550792257 ----- 62.3%
33 ----- 0.365988324268489 ----- 63.4%
34 ----- 0.355008674540435 ----- 64.5%
35 ----- 0.344358414304222 ----- 65.6%
36 ----- 0.334027661875095 ----- 66.6%
37 ----- 0.324006832018842 ----- 67.6%
38 ----- 0.314286627058277 ----- 68.6%
39 ----- 0.304858028246529 ----- 69.5%
40 ----- 0.295712287399133 ----- 70.4%
41 ----- 0.286840918777159 ----- 71.3%
42 ----- 0.278235691213844 ----- 72.2%
43 ----- 0.269888620477429 ----- 73.0%
44 ----- 0.261791961863106 ----- 73.8%
45 ----- 0.253938203007213 ----- 74.6%
46 ----- 0.246320056916996 ----- 75.4%
47 ----- 0.238930455209486 ----- 76.1%
48 ----- 0.231762541553202 ----- 76.8%
49 ----- 0.224809665306606 ----- 77.5%
50 ----- 0.218065375347407 ----- 78.2%

[Heroic]

Significance of Findings
The difference between a sci civ and a non sci civ gets larger the more unique techs there are to research. After just 10 unique researched here is the comparison:

SCIENTIFIC ----- 40.1%
PRIMITIVE ----- 26.3%

This is now a 14.2% difference between a SCI vs PRIM civ. Lets translate this into real meaning. If you are the first to research 10 techs and you are primitive, you have about a 1 in 4 chance of an SGL. IF you are scientific though, your chances are much better than 1 in 4. Your chance are even much better than 1 in 3. Your chance are close to 2 in 5!! In a longer game, the scientific civ's odds get better and better than the primitive civ's odds. The difference increases, but it increases by less and less.

Real World Application
So.......how realistic is it to get 10 unique techs? The key is getting those unique techs. The civ that seems best qualified for the job would probably start with alphabet but also be scientific. That would mean.....GREECE. A likely path would likely be as follows:

1 ----- Writing
2 ----- Philo
3 ----- MapMaking (free from Philo assuming you traded for pottery)
4 ----- Literature
5 ----- Code of Laws
6 ----- Republic
7 ----- Free tech for advancing ages
8 ----- other
9 ----- other
10 ----- other

Doable in a decent length game - certainly in a long game. Even if you only got 7 techs, a sci civ has about a 1 in 3 while a prim civ has about a 1 in 5 chance.

Final Notes
SCI makes a reasonable difference even in games that stay ancient ( providing a 1 in 3 chance which is better than the normal 1 in 5 ). However, SCI will show itself better the longer the game is. Do remember though, that these odds do not guarantee a SGL. If you follow the tables to the bottom you will notice that the odds get better by a smaller and smaller amount but never quite reach 100%. Alternatively, you could luck out with more than 1 SGL...but those would be very very small odds for both SCI and PRIM....enough that it is not likely to be a noticeable difference. Also, bear in mind that timing can be crucial.....if that SGL comes on your last turn, your 1 in 3 may prove itself true.......but how much good is a last turn wonder? Well.....now that we have polished our abacus........have fun and go discover wonderous new technologies or see about perfecting the invention of the Chasqui Tutu.

-Heroic

[FriedrichPsitalon]

Excellent, groundbreaking research and explanation, complete with a thoroughly accurate assessment of how to use this information.

Top notch, worthy of the sticky this thread is about to get; I strongly reccomend you post this in Apolyton's strategy forum as well, since even the SP community will find it useful, though their strategy for which research when may be different since they know the AI's habits.

Two thumbs up!

[Mo D]

I realized the power of that extra 2% right when C3C came out. I am definitely a fan of the scientific trait - especially in epics.

One thing that's important to note: most players say "5% vs 3%? Who cares?" That's not the way to look at it. It's not a 2% difference between scientific and primitive. If you are a scientific civ, you have a whopping 66% better chance of generating an SGL from any one tech than does your primitive opponent.

Scientific is pretty nice when you think of it that way, eh? ;D

Nice job Hero. SGL's are cool.

[Nickel]

Man thats really good to know heroic, and thanks for the tables.

[Hrathnir]

I thought about huts before as opposed to loss of chance for sgl.

My rational is that unless you are way ahead of opponent in techs its always good to grab the hut for a chance at a free tech to stay ahead of the opponent for more chances later. If opponent catches up then you get less chances as the game moves forward.

[cmdishr]

Wow hero....nice job bud!! one thing however, I dont think the hut thing is quite that simple eventhough it does decrease the amount of techs researched doesnt it increase the chance of reasearching a unique tech since it moves you along the tech tree? Would really have an impact in a game including more than 1 scientific civ.

[Heroic]

Glad you all like this one :-) Thanks for the support y'all :-). I also agree with you Argh. Huts are the enemy.....but kind of an unavoidable enemy. If you dont get the hut....someone else probably will so it still hurts you by them getting techs to catch up or even pass you. Also, completely true that you should do the huts and if the tech comes it either gives tech that allows new uniques....or it may allow you to trade for tech that will allow new uniques. So as cmdishr notes, it can help get a chance at unique techs. The only other sad but inevitable thing about it is that the techs further down the tree cost more to research....so the huts will slow down the process some. Good Considerations.

-Heroic

[lastride63]

Ok, I may be wrong but I thought that to get an SGL, you had to be in a tech "race". I mean race as you and another player are researching the same tech that no one else has and you beat them to it? Like I said I may be wrong but it seems that is the way it works, to me anyway

[CrudeDirk]

Personally I'm a follower of Bool's law which is something like this... 'All variables are reset after each event' Which means the odds at any time are always. the same, always a 5% chance for a sci civ.

So how does this affect your thoughts?

Another way to look at this is no matter how many weeeks you buy a lotto ticket the odds are always the same, always a 1 in a million chance or whatever.

[donaldkipper]

yes but if you do something many times, the odds of something occuring one of those many, or every time, change

take tossing a coin, there is 50% chance of getting heads or tails
if i toss it again, forgetting the last toss, once again i have a 50% chance of getting heads or tails (this is the point u just made)

however, if u say i am going to toss a coin twice, what is the chance i will get heads at least once, then u get new odds - the more times you toss the coin hoping for heads once, the better chance you have of being successfull.

in the case of SGL's. the discovery of each tech holds a 5% chance (for sci civs), but if u discover 3 unique techs, u have a better than 5% chance of getting one SGL.

[Don K Hotay]

Oh lord, the binomial distribution....reading your calculations Heroic (during which I fell asleep ), I felt like I was back in probability class, where I also fell asleep many times.

[Mo D]

Stats class was a b****.