Interesting game : 9 sixes in a row

[Ankit91]

An interesting game. 9 sixes in a row. Luckily, I was on right side of variance giving me 18 ores and around 20-22 sheep.
https://boardgamearena.com/table?table=552355606

[Mathew5000]

9 sixes in a row.

Claim verified. Starting at move 129: https://boardgamearena.com/gamereview?table=552355606

[Stroom]

This can happen. If there are hundreds of games played every day, these things are going to happen eventually. If you have a game of 72 moves, you can have any combination of dice rolls from 72 x 2 to 72 x 12. Most of the time it would be 2x2, 4x3, 6x4, 8x5, 10x6, 12x7, 10x8, 8x9, 6x10, 4x11, 2x12. But there are just so many other combinations that are fair or unfair... I'd say that for about 50% of the combinations a human would claim to be "unfair". So either just deal with it or play a different game.

[euklid314]

This can happen. If there are hundreds of games played every day, these things are going to happen eventually. If you have a game of 72 moves, you can have any combination of dice rolls from 72 x 2 to 72 x 12. Most of the time it would be 2x2, 4x3, 6x4, 8x5, 10x6, 12x7, 10x8, 8x9, 6x10, 4x11, 2x12. But there are just so many other combinations that are fair or unfair... I'd say that for about 50% of the combinations a human would claim to be "unfair". So either just deal with it or play a different game.

I have to correct you, Stroom. It is extremely(!) unlikely that 72 dice throws result in 2x2, 4x3, 6x4, 8x5, 10x6, 12x7, 10x8, 8x9, 6x10, 4x11, 2x12 or anything even close to such a distribution!

I am well aware that you probably mean it correctly since you mention all the other "unfair" combinations that are possible. But many people are not aware that almost every(!) experiment of 72 dice throws must(!) have several numbers that appear much too often or much too rare. Just because 72 rolls are very few and the perfect "random" distribution is only (roughly) approached after many 1000s of rolls.

Just consider a sequence of 72 rolls and stop after, say, 36 rolls. Only if the dice had memory the next 36 rolls could approach the "expected" distribution for 72 rolls. It is extremely unlikely that the later 36 rolls are approximately opposite to the first 36 rolls. Why should they? The dice have no memory and there is no reason to believe that the numbers that were rare in the first 36 rolls should appear more often in the next 36 rolls and vice versa.

Please note that for a very large number of experiments the "expected" distribution is *not* reached because those numbers that were rare in the beginning appear more often later: If you first throw a coin for 10 times and have only 3 heads and then you throw it another 90 times it is not necessary that heads will be overrepresented in those following rolls. Even if you roll 35 times heads out of 90 (i.e. again heads is underrepresented), the proportion tends towards the expected 0.5 ratio. 38/100 is closer to 0.5 than 3/10.

I repeat myself: Dice have no memory and still: On a large scale the expected distribution will be approxmated better than at a small scale. 72 rolls is a *very* small scale.

[Mathew5000]

It's worth pointing out that nobody in this thread called the particular streak "unfair". OP just described it as "interesting". Out of curiosity I ran a Monte Carlo simulation of 10,000,000 trials. In each trial, a pair of dice was thrown 51 times (the same number as in the game OP linked to). In 55 of the trials, there was a run of 9 or more of the same value. Full results:

Code: Select all

Number of trials in which the longest run was...
11 or more:          0
10:                  5
9:                  50
8:                 293
7:               1,993
6:              13,785
5:              94,629
4:             662,922
3:           3,880,410
2:           5,318,641
1:              27,272

total:      10,000,000

In trial #2,312,765 there was a run of 9 fives. Here are the rolls from that trial:

Code: Select all

   [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
d1    5    1    2    3    5    5    5    1    4     6     2
d2    2    1    5    3    1    5    5    6    5     6     5
   [,12] [,13] [,14] [,15] [,16] [,17] [,18] [,19] [,20]
d1     2     5     2     4     2     5     6     6     5
d2     2     3     6     3     6     5     6     2     1
   [,21] [,22] [,23] [,24] [,25] [,26] [,27] [,28] [,29]
d1     1     6     2     3     1     4     6     3     6
d2     2     6     2     5     2     2     3     1     1
   [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
d1     5     5     5     4     5     6     4     1     3
d2     1     5     3     3     1     2     3     4     5
   [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47]
d1     1     5     1     4     3     2     2     3     4
d2     3     6     5     1     2     3     3     2     1
   [,48] [,49] [,50] [,51]
d1     4     3     4     2
d2     1     2     1     6

In trial #6,673,240 there was a run of 9 nines.

Other than those two trials, every time there was a run of 9 or more, the value was six, seven, or eight. Seven times it was six; 35 times it was seven; and 11 times it was eight.

[Jellby]

How do you prove your Monte-Carlo simulation is unbiased?

[Stroom]

I have to correct you, Stroom. It is extremely(!) unlikely that 72 dice throws result in 2x2, 4x3, 6x4, 8x5, 10x6, 12x7, 10x8, 8x9, 6x10, 4x11, 2x12 or anything even close to such a distribution!

Yes, that's right. My mistake. It is what theoretically happens most often... but even then it is very rare.

[Ankit91]

This can happen. If there are hundreds of games played every day, these things are going to happen eventually. If you have a game of 72 moves, you can have any combination of dice rolls from 72 x 2 to 72 x 12. Most of the time it would be 2x2, 4x3, 6x4, 8x5, 10x6, 12x7, 10x8, 8x9, 6x10, 4x11, 2x12. But there are just so many other combinations that are fair or unfair... I'd say that for about 50% of the combinations a human would claim to be "unfair". So either just deal with it or play a different game.

I never claimed game to be unfair. I fully understand variance. However unlikely a sample is, it can still happen. That is probability for you.

[Ankit91]

It's worth pointing out that nobody in this thread called the particular streak "unfair". OP just described it as "interesting". Out of curiosity I ran a Monte Carlo simulation of 10,000,000 trials. In each trial, a pair of dice was thrown 51 times (the same number as in the game OP linked to). In 55 of the trials, there was a run of 9 or more of the same value. Full results:

Code: Select all

Number of trials in which the longest run was...
11 or more:          0
10:                  5
9:                  50
8:                 293
7:               1,993
6:              13,785
5:              94,629
4:             662,922
3:           3,880,410
2:           5,318,641
1:              27,272

total:      10,000,000

In trial #2,312,765 there was a run of 9 fives. Here are the rolls from that trial:

Code: Select all

   [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
d1    5    1    2    3    5    5    5    1    4     6     2
d2    2    1    5    3    1    5    5    6    5     6     5
   [,12] [,13] [,14] [,15] [,16] [,17] [,18] [,19] [,20]
d1     2     5     2     4     2     5     6     6     5
d2     2     3     6     3     6     5     6     2     1
   [,21] [,22] [,23] [,24] [,25] [,26] [,27] [,28] [,29]
d1     1     6     2     3     1     4     6     3     6
d2     2     6     2     5     2     2     3     1     1
   [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
d1     5     5     5     4     5     6     4     1     3
d2     1     5     3     3     1     2     3     4     5
   [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47]
d1     1     5     1     4     3     2     2     3     4
d2     3     6     5     1     2     3     3     2     1
   [,48] [,49] [,50] [,51]
d1     4     3     4     2
d2     1     2     1     6

In trial #6,673,240 there was a run of 9 nines.

Other than those two trials, every time there was a run of 9 or more, the value was six, seven, or eight. Seven times it was six; 35 times it was seven; and 11 times it was eight.

Nice work Matthew.
Did some rough calculation to match your Monte Carli simulation results. No
Mathematically, for 7 , probability for 9 consecutive rolls= 1/6^9=9.92 ×10^-8
Since 51 throws, 43 such 9 paired sequences. But there will be overlap and double counting.
So, probability should be less than 43× 9.92 * 10^-8=0.0000042. So, your probability 0.0000035 seems about right.
Similarly for 6,8 = 5/36^9×43 =0.00000082. Again quite close to your numbers.

[Jontia]

Main Catan page shows 3.6 million games of Catan have been played on the site. So seems pretty likely there absolutely should be very lopsided dice results in that history log.