ZERO fives two games in a row

[Mathew5000]

I recently played a game with a very low number of sixes thrown and won as 23 4s were thrown and I was on 4s. Next game I thought that can't happen again so placed two settlements on sixes.

https://en.wikipedia.org/wiki/Gambler%27s_fallacy

[Pablo Escobear]

I recently had a game with 22 5's and 22 9"s and 0 (zero!) 6's or 4's. But 10 3's and 10 11's..

Also: from my last 10 games the number 9 was rolled 16% (normally it should be 11%). I understand it can happen in a game, but in 10 games that average..

[euklid314]

I recently had a game with 22 5's and 22 9"s and 0 (zero!) 6's or 4's. But 10 3's and 10 11's..

Also: from my last 10 games the number 9 was rolled 16% (normally it should be 11%). I understand it can happen in a game, but in 10 games that average..

You wanted to say 22% 5s and 22% 9s (game %417465500). That is quite a lot.

But let us look closer at your numbers: In your last 10 games (newest game on top) - where I will ignore those games that had 0 rolls - you truely had a lot of 9s:

Number of 9s Number of rolls Percentage of 9s (rounded to full integers, similar as BGA does)
8 52 15%
14 76 18%
(0 0 NaN)
9 41 22%
9 53 17%
(0 0 NaN)
6 66 9%
9 53 17%
10 53 19%
8 59 14%
10 67 15%
9 92 10%

Total:
92 612 15.03%

Thus, in 612 rolls you had 92 9s (instead of the "expected" 68 9s), which is 15.03% instead of the expected 11.11%.

The probability that you get 92 or more 9s in 612 games is 0.0018 = 0.18%, i.e. a probability of 1 in 555.

My Analysis: If 555 random players *only* look at their last 10 games (!) and *only* look at the number 9 (!), one of them is expected to have your high percentage of 9s (or even higher!).

One other player of the same 555 players is expected to have 92 or more 5s (same probability of 0.18%). He will be equally surprised. One other of these 555 players is expected to only have rolled 46 or less 9s (also probability of 0.18%). He will be "shocked" about rolling so few 9s! And then there is one person beeing surprised about rolling only 46 or less 5s. We have 4 surprised people already. If we look at all possible numbers 2,3,...,12 and not only at 5s and 9s we will have approx. 22 out of 555 players that will be surprised, same as Pablo Escobear.

Please note that we have only looked at the last 10 games of each player. If we look further back in the game history (we humans have surprisingly good memory of surprising facts) then in the 481 games of CATAN of Pablo Escobear alone there are 472 different sets of 10 consecutive games. There is almost a guarantee that in this playing history there have happened several such events as rolling 92 9's out of 612 rolls.

Nevertheless this is truely a rare event which one will most likely take notice of. I hope I could make clear that "rare event" is not the same as "unlikely event" for this special case.

[nsdbs]

dice rolls.jpg (20.86 KiB) Viewed 727 times

I know it's not zero 5s but, WTH? What's up with the algo?

[Jellby]

Nothing. 55 rolls are just not enough to draw any kind of statistics, it's not even twice the "naïve minimum" of 36. Displaying percentages does more harm than good.

[euklid314]

The game had only 53 rolls and it had 12 4s.

The probability of 12 or more 4s is 0.1169% or approx. 1 out of 855.

If everybody posts each single game event with a probability of <1:800, there will be many hundreds (thousands?) of forum posts every day (only for CATAN). Just because there are *many* patterns that we humans do spot and there are many players and many games.


The previous poster at least posted a 1 out of 550 event that went over 10 games and 612 rolls. And even then a mathematician will not be impressed - as I tried to make clear in my response.

[Cos-]

Please note that we have only looked at the last 10 games of each player. If we look further back in the game history (we humans have surprisingly good memory of surprising facts) then in the 481 games of CATAN of Pablo Escobear alone there are 472 different sets of 10 consecutive games. There is almost a guarantee that in this playing history there have happened several such events as rolling 92 9's out of 612 rolls.

Nevertheless this is truely a rare event which one will most likely take notice of. I hope I could make clear that "rare event" is not the same as "unlikely event" for this special case.

I think your final couple of sentences don't sum this up well. It would be more accurate to say that this is a rare event that people will most likely not notice. Also, it is rare precisely because it IS unlikely. What you're missing, or not saying, is that nearly every possible sequence of roll frequencies is unlikely, and there are a lot more truly rare sequences of rolls frequencies than there are ones that aren't so rare. So while the odds of this particular "rare" event were very low, the odds that some rare event was going to happen are always extremely high.

Pretend, for example, that you're driving on a major long distance road and decide that at 11:00am you will look at the first car you see and note its license plate. You wait until 11:00, look, and see license plate "46Z 780R". Well, what are the odds that the one license you would see at 11:00 on that road that day would be that one, "46Z 780R"? You certainly could not have predicted that, nor could anyone else! It's a very very unlikely thing. It could have been any one of so many other license plates, and yet it turned out to be that one! Wow.

But you could say the same about any car's plate you might have seen. They're all rare and unlikely. Whichever one you happen to see, it's nearly certain that it was going to be one that is rare and unlikely. That, you could have predicted.

[Pablo Escobear]

Thanks for the mathematically insight. I agree that it's all about chances and eventually it will all even out.
It's just that rare events in a short period stick in your memory. If it was more spread out, I would never have noticed it.

It keeps going though, here are my last 5 games (games not played not counted):
Number of 9s Number of rolls Percentage of 9s (rounded to full integers, similar as BGA does)
10 52 19%
5 47 11%
10 51 20%
14 60 23%
7 63 11%

Total:
46 273 16,84%

[Jellby]

I agree that it's all about chances and eventually it will all even out.

Just a sidenote. In the long run it will even out. But that does not mean, as we all tend to think, that in the future you will more 5s than expected (or whatever is needed to "even out" the statistics). What it means is that whatever deviations from the expected probabilities will be less and less noticeable in relative terms, but not in absolute terms.

As a example with a coin toss. If you toss a coin 100 times, you're unlikely to get 50 heads and 50 tails. Let's say you got 45 heads and 55 tails. Now if you toss it 900 more times (for a total of 1000), you're not going to get 455 heads to "compensate" for the low previous rate. In fact, it could very well be (50% chance) that you again get more tails than heads. But even if you again get 10 more tails than heads, the total will be 490 vs 510. That is an even larger absolute deviation than with 100 tosses (20 vs 10), but it's a smaller relative deviation (2% vs 10%). This is what you can expect.

[Cos-]

Thanks for the mathematically insight. I agree that it's all about chances and eventually it will all even out.
It's just that rare events in a short period stick in your memory. If it was more spread out, I would never have noticed it.

It's not really that rare events stick in your memory. You're constantly experiencing a lot of rare events and forget most of them quickly. Most roll patterns you see in Catan games are rare. What sticks in your memory are events that affect you significantly, or that have a pattern-like look to them.

For example, I think the probability of rolling no fives at all in 20 consecutive rolls of two dice is around 9.5%, while the probability of rolling a 5 exactly four times in 20 consecutive rolls is around 11% - not much different. They're both similarly "rare" events. But if you have a settlement on a 5 and 5 doesn't get rolled for 20 consecutive rolls, that's going to stick in your mind. On the other hand, if 5 comes up four times in 20 consecutive rolls in the game, that will probably not be particularly memorable to you. If you think about it, whether 5 comes up three, four, or five times won't make as big an impression on you - those all "feel" like pretty similar levels of rolling a 5. But 0, that feels very different from all others.