Recommend
6 
 Thumb up
 Hide
13 Posts

Terra Mystica» Forums » General

Subject: Spade scoring tile (warning - math required) rss

Your Tags: Add tags
Popular Tags: [View All]
Edward Reece
United States
Garland
Texas
flag msg tools
Chance of pulling any single scoring tile in Terra Mystica.

We have 8 tiles, tile #1 is the "spade" tile.
The spade tile cannot be chosen for the 5th or 6th round.
The other 7 tiles all have the same probability of showing up in any given game.

The probability of a tile being chosen on the first draw is 1/7 (because the spade tile is not available to be drawn)
That is 14.29%

The probability of a tile being drawn for the next spot is 1/6 (spade tile is still not available and some other tile has been removed)
That is 16.67% but it is 16.67% of the 85.71% of the time that the tile was not drawn the first time, that is 14.29%.

The probability of a tile being drawn for the next spot is 1/6 (spade tile is now available and two other tiles have been previously removed)
That is 16.67% but it is 16.67% of the (100 -14.29 -14.29)% of the time that the tile was not drawn previously, which is 11.91%.

Continuing on we get
1/5 giving you 20% of (100 -14.29 -14.29 -11.91)% of the time that the tile was not drawn previously, which is 11.90%
1/4 giving you 25% of (100 -14.29 -14.29 -11.91 -11.90)% of the time that the tile was not drawn previously, which is 11.9%
1/3 giving you 33% of (100 -14.29 -14.29 -11.91 -11.90 -11.90)% of the time that the tile was not drawn previously, which is 11.78%

So to recap -
In 14.29% of the games tile X (X being any tile except the spade tile) will be drawn first (and placed on the round 6 slot)
In 14.29% of the games tile X will be drawn on the second draw (and placed on the round 5 slot)
In 11.91% of the games tile X will be drawn on the third draw (and placed on the round 4 slot)
In 11.90% of the games tile X will be drawn on the fourth draw (and placed on the round 3 slot)
In 11.90% of the games tile X will be drawn on the fifth draw (and placed on the round 2 slot)
In 11.78% of the games tile X will be drawn on the sixth draw (and placed on the round 1 slot)

Meaning that tile X will be drawn at some point in 76.07% of the games. This is of course the same for all scoring tiles except the spade tile.

When you perform the same calculation for the spade scoring tile
1/6 = 16.67%
1/5 = 20% (100 -16.67)% = 16.67% chance of being drawn
1/4 = 25% (100 -16.67 -16.67)% = 16.67% chance of being drawn
1/3 = 33% (100 -16.67 -16.67 -16.67)% = 16.5% chance of being drawn

The spade scoring tile will now only be in 66.51% of games.


This means the spade scoring tile shows up in noticeably fewer games than any other single tile, and it does so because of the manner of choosing tiles.

A simple fix:
Draw six tiles - these are the tiles you play with, after that then order them as usual. Then all tiles have the same chance of being drawn to be "in the game".
6 
 Thumb up
1.00
 tip
 Hide
  • [+] Dice rolls
Evil Roy
United Kingdom
Sutton
Surrey
flag msg tools
Avatar
mbmbmbmbmb
There's something wrong with your maths. The probability of any scoring tile appearing in round 4 is the same as the same tile appearing in round 1.
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Evil Roy
United Kingdom
Sutton
Surrey
flag msg tools
Avatar
mbmbmbmbmb
EdFactor wrote:
The spade scoring tile will now only be in 66.51% of games.


Once the tiles for rounds 5 and 6 have been selected you have 6 left. Four out of these six will be chosen for the first four rounds so the chance of the spade tile appearing in 4/6 or 2/3 (~66.67%).
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Evil Roy
United Kingdom
Sutton
Surrey
flag msg tools
Avatar
mbmbmbmbmb
The chances of a non-spade scoring tile coming out are 16/21 (~76.19%). You can check this as the total of the probabilities should total 6 (as 6 of the tiles will be drawn).
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Shawn Fox
United States
Richardson
Texas
flag msg tools
Question everything
Avatar
mbmbmbmbmb
Are you playing with house rules here? There is no rule stating that the spade scoring tile can't be used in the 5th or 6th round so far as I am aware. Did I miss some 2nd edition rules or something?
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Thomas Leitner
United States
Madison
Wisconsin
flag msg tools
badge
Avatar
mbmbmbmbmb
sfox wrote:
Are you playing with house rules here? There is no rule stating that the spade scoring tile can't be used in the 5th or 6th round so far as I am aware. Did I miss some 2nd edition rules or something?

"The official setup of the scoring cards now reads:

The Game board displays 6 spaces for the Scoring tiles (numbered 1-6). Each such space corresponds to one round. Shuffle the Scoring tiles and place them face-up on these spaces one after another, beginning with space 6. If you draw the Scoring tile with a Spade on its left side for spaces 5 or 6, put it aside and draw another one instead. Then shuffle it back in. (In the example, this tile is on space 3). Finally, cover the right half of the Scoring tile on space 6 with the Game end token.


The reason for this update is the danger of an exploit of the transforming bonus as our tests are showing.

Unfortunately we are in the middle of the next printrun and we can not risk to have it in the rules, so as a precaution we will ban the spade scoring tile from round 5 and 6. If you do not see the issue here feel free to ignore it in your house rules. But the official rules have to be on the save side..."

Quoted from
Frank Heeren
Germany
Eppstein-Bremthal
flag msg tools
publisher
badge
Avatar
mbmbmb
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Edward Reece
United States
Garland
Texas
flag msg tools
Evil Roy wrote:

Once the tiles for rounds 5 and 6 have been selected you have 6 left. Four out of these six will be chosen for the first four rounds so the chance of the spade tile appearing in 4/6 or 2/3 (~66.67%).


Evil Roy wrote:

The chances of a non-spade scoring tile coming out are 16/21 (~76.19%). You can check this as the total of the probabilities should total 6 (as 6 of the tiles will be drawn).



Ok if my math is a little off, your math still supports my thesis that the spade scoring tile will be appearing in significantly fewer games then any of the other tiles. (about 9.5% less then any other single tile)
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Evil Roy
United Kingdom
Sutton
Surrey
flag msg tools
Avatar
mbmbmbmbmb
EdFactor wrote:
Ok if my math is a little off, your math still supports my thesis that the spade scoring tile will be appearing in significantly fewer games then any of the other tiles. (about 9.5% less then any other single tile)


Yes, I think it's probably just rounding errors.
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Edward Reece
United States
Garland
Texas
flag msg tools
MDJD wrote:
sfox wrote:
Are you playing with house rules here? There is no rule stating that the spade scoring tile can't be used in the 5th or 6th round so far as I am aware. Did I miss some 2nd edition rules or something?

"The official setup of the scoring cards now reads:

The Game board displays 6 spaces for the Scoring tiles (numbered 1-6). Each such space corresponds to one round. Shuffle the Scoring tiles and place them face-up on these spaces one after another, beginning with space 6. If you draw the Scoring tile with a Spade on its left side for spaces 5 or 6, put it aside and draw another one instead. Then shuffle it back in. (In the example, this tile is on space 3). Finally, cover the right half of the Scoring tile on space 6 with the Game end token.


The reason for this update is the danger of an exploit of the transforming bonus as our tests are showing.

Unfortunately we are in the middle of the next printrun and we can not risk to have it in the rules, so as a precaution we will ban the spade scoring tile from round 5 and 6. If you do not see the issue here feel free to ignore it in your house rules. But the official rules have to be on the save side..."

Quoted from
Frank Heeren
Germany
Eppstein-Bremthal
flag msg tools
publisher
badge
Avatar
mbmbmb


I am only suggestion that there is a different way to do the draw that doesn't reduce the probability that the spade scoring tile will come out in any given game.
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Tables
United Kingdom
Coventry
flag msg tools
badge
Avatar
mbmbmbmbmb
As Roy points out, you have a lot of rounding errors. You should always be doing your maths in fractions, and convert at the end.

Not to mention, the conclusion was unsurprising, to say the least.

The spade bonus tiles have a 1/6 chance of appearing on spaces 1, 2, 3 and 4 and 0 chance of appearing on spaces 5 or 6.
The other tiles each therefore have a 1/7 chance of appearing on spaces 5 or 6 and a 5/42* chance of appearing on spaces 1, 2, 3 and 4.

*There's a 1/6 chance of it having a spade, so 5/6 over 7 tiles = 5/6*1/7 = 5/42

The chance of having a spade scoring tile in the game is 2/3. The chance for any other tile is 16/21. In percentages, that's about 66.67% to 76.19%
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Gareth
Switzerland
Zürich
Zürich
flag msg tools
badge
My ass is green, but I don't let it worry me.
Avatar
mbmbmbmbmb
EdFactor wrote:
A simple fix:
Draw six tiles - these are the tiles you play with, after that then order them as usual. Then all tiles have the same chance of being drawn to be "in the game".


Indeed, this is a nice simple method which fixes the issue.
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Cooper Matlock
United States
Texas
flag msg tools
Nice post, Edfactor. Thanks for contributing.
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Front Page | Welcome | Contact | Privacy Policy | Terms of Service | Advertise | Support BGG | Feeds RSS
Geekdo, BoardGameGeek, the Geekdo logo, and the BoardGameGeek logo are trademarks of BoardGameGeek, LLC.