More On Roll Probability

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andycupid
Posts: 48
Joined: 31 July 2015, 14:22

More On Roll Probability

Post by andycupid »

So I've been playing this game a fair amount lately. My results were pretty decent, but up and down. At best I was top 10 in elo at 430ish and first in arena, at worst I was 170 and 2000+. (That was noted just so that my claims won't be classified as ranting just because I'm bad lol) Sometimes this game can be frustrating, but then I knew a clever selection of move choices can win most games of even luck (mine is 60~70%).
Now everyone has good and bad days, but browsing through a particular forum discussion about the source code had me started observing what 4 die I roll each round. While I don't have any statistical approach by any means, nor do I have even the slightest understanding of coding, I do think that three of a kind is being rolled way more often than its calculated probability indicates. As well as the chances of rolling two pairs. Well, actually, I see those two situations appear more than rolling 4 completely different numbers (which isn't easy, but should be easier than those two). Sure, I can very well ignore that, but then I believe an uneven distribution on probability makes games slightly more in favour to overly cautious players. i.e. In a game where I somehow keep rolling three of a kinds or pairs consisting of 1s or 6s (see, they happen more than I would like), that makes it hard to make a comeback even when my opponent is turtling his way through. At the end of the day, I don't really mind losing, but as I look to play competitively, I do mind losing randomly to unrefined strategies.

To further note, while I was on a terrible streak, I experimented with the rule posted here: http://www.taterenner.com/cantstop.php
His playing method is based on expected loss and expected gain, which not only factors probability alone but also multiplied by profit and turns made. That's the same as the casino rule, meaning that following such rules in the long run should result in a net gain, and that's why casinos and banks and the lottery hardly ever lose money. So if you take too little moves, you're making undervalue plays, and you should be losing more than you win. If you go too far, then in the long run you should be failing to win as well. I tried following the strictly in the first 2 to 3 rounds exclusively when there isn't pressure from the opponent, and, well got stuck in 200 elo and 1450+ EAS. Then I reverted back to my unanalysed strategy that brought me to top 50 and guess what, that works. (Of course I'm not writing down my climbing strategy lol.)

What I wanted to ask was, has anyone observed such patterns in dice rolling? Obviously the most important thing is to have fun, but a bit of discussion to become better can't be distasteful...right?
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Jest Phulin
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Joined: 08 July 2013, 21:50

Re: More On Roll Probability

Post by Jest Phulin »

andycupid wrote: 05 June 2020, 02:58 While I don't have any statistical approach by any means,
Any discussion about die rolls without a statistical approach will be met with high resistance. The human mind is very poor at determining randomness. The only way to analyze it is with a statistical approach.

There are 1296 ways of rolling 4 dice (6*6*6*6). It's late at night and I'm having trouble with my probabilities, but I can create a spreadsheet to create every dice roll, and analyze it for unique numbers, 3-of-a-kind, and 2-pairs.
andycupid wrote: 05 June 2020, 02:58 I do think that three of a kind is being rolled way more often than its calculated probability indicates.
This comes up 120 times. in 1296 rolls, or about 1 in 10 rolls. The average game on BGA is a little over 30 rolls per player, which means that triples coming up for you three times a game is expected.

--edit While some sites show this probability of (1)*(1/6)*(1/6)*(5/6) (die A can be anything, B matches A, C matches A, D does not match, or 5/216), this fails to take into consideration that 1-1-1-2 is a different roll than 2-1-1-1, which is different than 1-2-1-1 which is different than 1-1-2-1. Since the odd die (the 5/6 probability) can be any one of the 4 positions, the 5/216 needs to be multiplied by 4, giving 20/216 (or 120/1296) end edit-----
andycupid wrote: 05 June 2020, 02:58 As well as the chances of rolling two pairs.
This is 90 times, or about 1 in 15. So twice a game. I'm too tired to do the actual ordering probability, sorry.
andycupid wrote: 05 June 2020, 02:58 Well, actually, I see those two situations appear more than rolling 4 completely different numbers (which isn't easy, but should be easier than those two).
This is 360 of 1296 times, or less than 1 in 4 times. But this is where the human brain isn't good at probability. Intuitively, the average person will believe that 4 dice should create 4 unique numbers, not a triple or 2 pairs. So it remembers the triple and double pairs rolls more than it remembers the unique rolls.

So, your homework: Replay 5 random 3-player games. Count 4 items in the games. A) the number of rolls. B) the number of triples. C) the number of 2 pairs. D) the number of no pairs (all unique).
Assuming that A is around 450, B should be about 45, C should be about 30, and D should be about 125. Only if the numbers are significantly different than this can the discussion proceed meaningfully. Otherwise, the game is behaving as expected -- randomly.



-----Edit. Sorry, it was late and my first post on this had some math errors. Sorry for anyone who saw the original post.
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nmego
Posts: 360
Joined: 27 December 2017, 07:08

Re: More On Roll Probability

Post by nmego »

I picked a random game, and did just that:
https://pastebin.com/BbYxTfCs
--------------------
All uniques: 38
1 Pair: 71
2 Pairs: 12
1 Triple: 11
1 Quadruple: 1
All moves: 133
--------------------
It took a good 20-30 minutes. I think the best way to do this is to just write a script that does for like 1000 games or so.

Interestingly, Uniques and Triples seem to fit their probability. It is hard to speak for Quadruple since it needs a much higher sample to calculate (Quadruples should have a 1/216 chance, a single game of 133 moves, or even 5 games are not enough).2 Pairs are a bit skewed (Showing up roughly 9%, while their probability should be 6.66%)
----------------------
Edit:
I also made a quick local test, rolling 4 dice 1e7 times and checking how much they pop up (Using python, which should use the same Mersenne Twister algorithm as the PHP random function)

Uniques: 2776937 (27.7%)
One Pairs: 5554232 (55.5%)
Two Pairs: 695964 (6.95%)
Triples: 927003 (9.2%)
Quadruples: 45864 (0.45%)
----------------------
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pkdubs
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Joined: 14 April 2020, 06:42

Re: More On Roll Probability

Post by pkdubs »

andycupid wrote: 05 June 2020, 02:58
What I wanted to ask was, has anyone observed such patterns in dice rolling? Obviously the most important thing is to have fun, but a bit of discussion to become better can't be distasteful...right?

I've been thinking about this type of strategy and I am wondering about the wisdom of the expected loss (x-) vs expected gains(x+) The problem, as I see it, is that if you always stop when x- exceeds x+, then you will sometimes be hit with a low-odds loss but you will never get the benefit of a low odds gain.

Example, with a three column start of 6 7 8, you should, on average, be able to roll 12 times before x- exceeds x+. The problem comes in the fact that, if you always stop at 12 rolls (feels pretty risky to go that far), you will sometimes get improbably wiped out at 2 consecutive rolls, but will never get the benefit of improbably rolling 20 consecutive rolls. Because 20 rolls will almost always win you a column, going for a more risky roll total seems to be a smart strategy.

Thoughts?
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SluggerBaloney
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Joined: 07 February 2020, 14:26

Re: More On Roll Probability

Post by SluggerBaloney »

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Jest Phulin
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Re: More On Roll Probability

Post by Jest Phulin »

SluggerBaloney wrote: 14 June 2020, 00:45
I do think that three of a kind is being rolled way more often than its calculated probability indicates
It is. I've done the analysis. Random probability says triples should come up around 1 in every 10-11 rolls. On BGA, triples come up on average 1 out of every 6-7 rolls. This was analyzed with a little over 1200 rolls.
Could you please submit a bug report (including which games were analyzed for the 1200 rolls) and post a link to it here? That way we can vote on it and get it corrected.
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SluggerBaloney
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Re: More On Roll Probability

Post by SluggerBaloney »

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Jest Phulin
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Re: More On Roll Probability

Post by Jest Phulin »

So, you are unable or unwilling to provide a list of games that you analyzed, so nobody else can verify. Fair enough...

Also, you seem to have an intolerance for the difference between a random number generated physically by the universe, and a pseudo-random number generated by a computer. For the purposes of this game (and all games, for that matter), a random number needs to provide two functions: 1) all possibilities need to happen with a defined probability (ie, if I roll a single 6-sided die, I will get each result 1/6th of the time [given a large enough time frame]), and 2) knowing the history of the previous numbers generated, it is impossible to predict the next number generated with a success rate better than a random guess.

The pseudo-number generator does both of these. And, if you really analyze physical dice (like Joseph Jagger did with roulette wheels at Monte Carlo in the late 1800s) you will find that rule 1 isn't followed.
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SluggerBaloney
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Re: More On Roll Probability

Post by SluggerBaloney »

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Tisaac
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Re: More On Roll Probability

Post by Tisaac »

SluggerBaloney wrote: 24 July 2020, 20:26 You are arguing from a point of belief, not data, so there's no point in bothering.
Yeah but you don't provide anymore data here so, as you said, "no point in bothering"...
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