SGFN-02: Spain to the Stars

According to my calculations the odds of all four civs getting the same tech is around 3.7% and all 3 techs being available between the four civs 44.4%.

The first calculation is easy, and Othniel had it almost right. It is 1/3^4 = 1/81 = 1.2% for all four civs to have one tech, but then you need to multiply by three, because there are three ways this could happen. So 1.2% is the odds of all 4 civs having a particular one of the three, and 3.7% is correct for all four civs to have exactly one of the three, but not any particular one.

For the second question, it is a bit tougher math. The first choice is irrelevant. You will get at least one of the three. On the second pick, you have a 1 in 3 chance of selecting the same tech, and a 2 in 3 chance of selecting something different.

First, take the case where the first two choices match. Now you must pick two different, non-matching choices with your next two picks. That's a 2/3 chance of a new tech on choice three, then a 1/3 chance on choice four of picking the last remaining tech. Therefore, total odds of this case = 1/3 * 2/3 * 1/3 = 2/27.

In the case where your first two picks do NOT match, you have a 1/3 chance to pick the last tech right off. In the 2 of 3 times when you fail, you then have a second 1/3 chance to pick the last tech. So that's 1/3 + 2/3*1/3 = 5/9. Times the 2/3 chance that we're in this situation to begin with, or 10/27.

The total answer is the combined totals of the two cases, or 12/27. That's a 44.4% chance, just as Thar concluded.

Ain't math great?
 
Ah yes, conditional probabilities. My fav. :D

Thanks for doing the math, anax. That's very interesting that there's a 44.4% chance of all three techs (if going into the Medieval or Industrial Ages ;)) showing up. Much higher than I would have guessed. Math is a beast.
 
Not really much of a bump- just 4 days. :)

I guess not. :)

I'm interested to see your calculations, especially on the 44.4% chance of all 3 techs being available. I rushed through my calcs so it's very possible I overlooked something. It'd be nice to have a pretty rock solid knowledge of the % chance for these events.

I made a program in C++ that simulates it around 30,000,000 times (which only takes a few seconds) and then just look at the results. I'm sure you can calculate it mathematically too but I don't know how, but I think it's kind of complicated. I mean the chance of getting three specific techs in a specific order is .33*.33*.33*.33 like you said I believe but there are many permutations and combinations that you have to sum up somehow.

Anyways, here are some results:

Chances of 3 techs being distributed between the teams:

3 teams:
1: 11.1%
2: 66.7%
3: 22.2%

4 teams:
1: 3.7%
2: 51.9%
3: 44.4%

5 teams:
1: 1.2%
2: 37.0%
3: 61.8%

6 teams:
1: 0.4%
2: 25.5%
3: 74.1%

7 teams:
1: 0.1%
2: 17.3%
3: 82.6%

Chances of 4 techs being distributed between the teams:

3 teams:
1: 6.2%
2: 56.3%
3: 37.5%
4: 0%

4 teams:
1: 1.6%
2: 32.8%
3: 56.2%
4: 9.4%

5 teams:
1: 0.4%
2: 17.6%
3: 58.6%
4: 23.4%

6 teams:
1: 0.1%
2: 9.1%
3: 52.7%
4: 38.1%

7 teams:
1: 0.0%
2: 4.6%
3: 44.1%
4: 51.3%

EDIT: And now I see anaxagoras's posted a great explanation. I haven't programmed for a while but it can be quite fun to just throw together a program if you see a problem like that. Anyways, great job with the game and I'm following the game with the Americans too.
 
Hi, is there still room for the roster? ^_^
I would like to join
 
This game's been over for quite a while now. Either lurk in some of the current SG's and see if they post any needs for more players, or wait until a new SG starts up (or start one yourself...).

SG's are a lot of fun, so I hope you find one to get involved with!
 
CBob03 and Slavemasters could both use another player...they're both in the winding-down, micromanagement-heavy stage, though.
 
Oh ok thanks ^.^ i'll try joining there
 
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