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I originally posted this article under my ladder name "Heroic" for the MP Ladder community. However since much of it applies to SP as well, I am posting it here too. For those interested, continuing MP discussion can be viewed at the following link http://civ3players.proboards2.com/in...325753&start=0
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!
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
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!
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
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