Number improbability

[Shazzypaz]

I checked the code for the "die appearance" statistics. They are incremented for the actual dice rolls and the sum of the two die, prior to any impact from any abilities that are used.

[Solarman9]

When you start the game with the 4 income card raised on first buy and the game decides you can not win. Is there any sort of hint that shows you when the algorithm has decided you automatically lose? the only number that rolled less then 4 is 10.

Die Appearances of 1
9
Die Appearances of 2
8
Die Appearances of 3
11
Die Appearances of 4
3
Die Appearances of 5
17
Die Appearances of 6
30
Die Appearances of 7
4
Die Appearances of 8
5
Die Appearances of 9
3
Die Appearances of 10
1
Die Appearances of 11
4
Die Appearances of 12
4

[King_Taravangian]

I have attached an image showing the actual probabilities for SpaceBase. If you can't see it, tell me what I did wrong and I'll upload it again

First clarification: The probabilities Solarman9 is talking about are for 2-dice-games where your only choice is to use the sum of the dice (Backgammon, for example). That is not true for SpaceBase, since you can choose individual dice instead of the sum (think about it, the smallest sum is 2, you would never activate sector 1 with those probabilities)

Second clarification: Even a sample of 100 is too small to validate probabilities. You can try this in excel, with the =RANDBETWEEN(1,2) function. Copy it over 100 rows and check the appearance rates, they are not 50%-50%. Or, if you have the time to do that, toss a coin 100 times and note the results.

I do not think the discussion should be deleted. It could be the case that other players are confused about the same thing and this might help shedding some light on the subject.

[Jellby]

What exactly is the "sum of dice + individual dice" probability measuring? It would be accurate if the mechanism was something like: Roll two dice, randomly select between "die 1", "die 2", "sum of dice". But in the actual game, the choice is not random, it depends on what was rolled and what your cards are. Not to mention that you don't choose between "die 1" and "die 2". If you want to measure something like how often you can choose some particular number, I think you still need to divide by 36, not by 108. The total sum would be > 100%, but that's fine, because the choices are not exclusive.

[King_Taravangian]

1. What is "Sum of Dice + Individual dice"

This refers to the columns in the possibilities matrix I considered.
Agreed, the choice is not random. But it also does not inflence other players, so if one rolls a 5 and a 2, a player can select 7, and another can choose 5 and 2, so all 3 (2,5 and 7) are valid appearances.

2. You don't choose between dies:

I didn't say you did, but if you rolled 3 and 3 the 3 appeared twice, the 6 appeared once. When I say "individual dice" I do not mean first or second die, I actually mean "Not the sum". So those individual appearances must be added to the pool of possibilities.

Whenever you compute the probability of something happening you use the formula Desired outcome/ Total possibilities. And the total possibilities are not 36, because those are just the combinations of two dice and do not consider "Not the sum" (to avoid saying individual dice). So you need to include the first 2 columns in the computation.

Probabilities not adding to 100% means that the denominator (Total possibilities) was calculated wrong, not considering certain possibilities.
These are probabilities for one player, where choices are exclusive. The actual appearances differ from player to player based on the choices made.

[King_Taravangian]

Let's take the 3 for example:
Chances in 2-dice-games are 5.6% and are calculated as 2/36. Why? Because there are only 2 scenarios where you can roll a three in those games (1-2 and 2-1).

However, in Spacebase you activate sector three even if you (or an opponent) rolls a 3-2. Because you do not choose the sum. So actually, there are a lot of other scenarios where you can activate sector 3 (marked in green). And you have to take these into consideration as well.

Please ask questions if this is still not clear. I am happy to answer.

[Jellby]

Probabilities not adding to 100% means that the denominator (Total possibilities) was calculated wrong, not considering certain possibilities.
These are probabilities for one player, where choices are exclusive. The actual appearances differ from player to player based on the choices made.

That's the point, they're not exclusive.

If one player chooses 3, that does not exclude the same player also choosing 5. If the roll is 3 and 5, the player can choose 3 and 5, so they're not exclusive.

Ignore the sums for a moment, and suppose you always take individual dice. The probabilities for all numbers should obviously all be equal, but they should not add up to 100%, because they're not exclusive.

[King_Taravangian]

Ignore the sums for a moment, and suppose you always take individual dice. The probabilities for all numbers should obviously all be equal, but they should not add up to 100%, because they're not exclusive.

I'm not sure I follow. If we ignore the sum, then the probability of each die appearing is 1/6; multiply that with 6 faces and you get 6/6, 100%.

If one player chooses 3, that does not exclude the same player also choosing 5. If the roll is 3 and 5, the player can choose 3 and 5, so they're not exclusive.

I see what you mean and I do not think we are in disagreement on this. In a roll of 3-5, there are three desireble outcomes (based on who you're asking): the 3, the 5 and the 8. They are presented in different cells, so all three are included in the Total possibilities. Each of the three is also included in the desirable outcome formula for that specific number. (you can see the 3 above)

For 8 it would be 5 scenarios/ 108 possibilities. Out of those 5, two are 5-3 and 3-5. So yes, they are not exclusive, but the formula accounts for this. That is why I have 3 columns as opposed to the one in Backgammon, to represent all the different choices you can make.

[King_Taravangian]

I color coded the table, so it would be more evident where each possibility is considered.
So, continuing the example of the 5-3 roll (criteria 27), I do the following:
1. Increment the numerator for all three (3, 5, 8) in their respective formulas
2. Consider these possibilities in the total (108).

Same goes for 3-5, which is a separate possibility, statistically speaking.
All posibilities are considered in one formula or another, so it naturally follows that they add up to 100%.

[Jellby]

I'm not sure I follow. If we ignore the sum, then the probability of each die appearing is 1/6; multiply that with 6 faces and you get 6/6, 100%.

Really? If I roll 2 dice, the probability that I'll see at least one 3 is 11/36: there are 11 possible outcomes with at least one 3 out of the 36 possible (and equally probable) rolls. The "average number of 3s" is, however, is 12/36 = 1/3: one of the 11 outcomes has two 3s. The "average proportion of 3s" is 12/72 = 1/6: each roll gives two numbers, not one.

For me, "the probability of each die appearing" means the first thing, the probability that I'll see at least one of that die, and if I sum all I get 66/36 > 100%, because in each roll I see two (possibly different) dice.