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Show HN: A Set of Dice That Follows the Gambler's Fallacy

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Re: Show HN: A Set of Dice That Follows the Gambler's Fallacy

#181

Post by andy_boot »

DINKDINK wrote: post #134
nickm12 wrote: post #73

I had the privilege of studying probability from G-C. Rota. One of my favorite quotes from him was "Randomness is not what we expect", which he used to describe the phenomenon of people disbelieving that random data was actually random. Another great was "This will become intuitive to you, once you adjust your intuition to the facts."

A good way to show how the "Randomness is not what we expect" phenomenon manifests is the following. Given a fair coin that outputs Heads (H) or Tails (T), which of the following sequences is more or less probable HHTHTT or HHHHHH Answer: Both are equally probable strings to be generated by the fair coin and each should occur once in every 64 sequences (~1.56%)

True But I would like to show you an interesting counterexample which requires changing the rules slightly. If we toss a coin and keep tossing until one of the below sequences turn up then which sequence is more likely?

HHHHHH THHHHH

Answer: THHHHH. Only 1/(2^6) of the time it will be HHHHHH. The only way to get HHHHHH will be if the first 6 flips are heads. If you don't flip a head you must have got a tail which means THHHHH will always appear before HHHHHH.

Re: Show HN: A Set of Dice That Follows the Gambler's Fallacy

#183

Post by zem »

reminds me of a very interesting demonstration from martin gardner. draw a 6x6 grid, and write a random digit in each cell, proceeding row by row. now count the number of pairs of consecutive (x, x) going horizontally versus vertically; you will almost always get doubled numbers in the columns because that's how random numbers work, but almost never in the rows, because when people are trying to generate "random" numbers by hand they avoid runs or other patterns.

Re: Show HN: A Set of Dice That Follows the Gambler's Fallacy

#185

Post by rcthompson »

koolba wrote: post #100
ideonexus wrote: post #8

A great application for this is in randomizing playlists. My friends, who are also CS grads and should know better, have often complained that their MP3 players, CD carousels, etc play the same music too often claiming that the random is broken, when a song repeating in a short period of time or other songs never playing is what you would expect from a truly random selection. Using this algorithm, you'd be sure to he…

A simpler way that's guaranteed to not repeat a song till the entire set has been played is: Assuming you have N songs, pick a prime P such that N is not divisible by P. Then pick a random index I (0 Another method is to pick a random key K and sort the entries of the playlist based on HMAC(SONG, K) (where SONG is the name or other identifier of the song). This has a number of interesting properties. For starters the…

I think you're solving the wrong problem here. The goal is not to take a list and shuffle it once. That's trivial. The goal is: given a set of N songs, sample an arbitrarily long sequence of songs with replacement from this set such that the sequence "feels" random. Things that are random but don't "feel random" include: playing the same song too often, not playing a song often enough, and repeating the same sub-sequence of songs too often. The "shuffle the list once and repeat forever" approach fails hard on the last criterion. Generating a new shuffle every time through the list solves that issue but fails on the first two criteria, since a song could appear at the end of one shuffle and at the beginning of the next, resulting in repeating the same song twice in a row (or vice versa, resulting in the song not playing for 2N-2 songs).

Re: Show HN: A Set of Dice That Follows the Gambler's Fallacy

#186

Post by hoytech »

acomjean wrote: post #106
Pxtl wrote: post #57

Earlier quoted context omitted.

You could just use a deck of cards with numbers on them. That's the simplest way to do "Gambler's Fallacy Dice" IRL. For example, there's a popular Catan expansion pack that replaces the dice with a deck of cards with numbers on them for just this reason, because in Catan each roll of 2D6 represents different regions of the map paying out so if a number never comes up it never pays out.

The thing about rolling 2D6, is the distribution isn't even. I'm not sure that factors into the game (have played but its been a while). Cards seem like a more controlled way to distribute the results with some randomness. only one way to get a 12 (6+6) but 7 much more likely.. (6+1, 5+2, 4+3, 3+4, 2+5, 1+6) as someone who played a bit of online backgammon back in the day, We always suspected the random dice rolls be…

As I'm sure you know, understanding this distribution is critical to good backgammon play. For example, all else being equal it's better to leave your exposed blots closer to a threatening opponent checker than further away (within 6 pips of course). Also, you need to account for rolling doubles which results in the counter-intuitive average pip value of a roll being 49/6 (~8.16).

An interesting bit of trivia: In roman times it seems they played with 3 dice instead of 2: http://www.bkgm.com/variants/Tabula.html

Re: Show HN: A Set of Dice That Follows the Gambler's Fallacy

#187

Post by andy_wrote »

brchr wrote: post #172
andy_wrote wrote: post #63

There's a probability model called the Pólya urn where you imagine an urns containing numbered balls (colored balls in a typical example, but to draw the comparison with dice we can say they're numbered 1-6), and every time you draw a ball of a certain color, you put back more balls according to some rule. A few probability distributions can be expressed in terms of a Pólya urn, see https://en.wikipedia.org/wiki/P%C3…

In the 1960s, the biostatistician Marvin Zelen proposed using something very much like the Pólya urn for clinical trials, calling it the "play the winner" rule [1]. This has had a major effect in causing a rethinking of the traditional randomized controlled trial, and these ideas are still making their way through the medical community today [2]. [1] https://www.jstor.org/stable/2283724 [2] https://www.fda.gov/downlo…

Interesting - just perusing those links, it sounds like a multi-armed bandit problem, in which you reason that if something has worked out before, you should tilt your bets more in that direction. In the context of the urn model, you'd return more balls of the same color for every successful draw. In the context of medicine, you can balance between proving or disproving a treatment effect and actually supplying that treatment to the test subjects who need them.

Relatedly, there's a Bayesian interpretation to overweighting successful past draws. A model where you return one extra ball of the same color to the urn gets you a Dirichlet-multinomial distribution, which is a die-roll distribution where the weights to each face are not known for sure, but are given a probability distribution and revised with observed evidence. In other words: here's an n-sided die, I don't know its weightings, but as I observe outcomes I'll update my beliefs that the sides that come up are more favorably weighted. The number of balls in the urn you start with correspond to your priors; only 1 ball of each color means a very weak belief that it's a fair die, 1000 balls of each color means a strong belief, unequal numbers mean that you start off believing it's weighted.

Re: Show HN: A Set of Dice That Follows the Gambler's Fallacy

#188

Post by bryanlarsen »

rcthompson wrote: post #185
koolba wrote: post #100

Earlier quoted context omitted.

A simpler way that's guaranteed to not repeat a song till the entire set has been played is: Assuming you have N songs, pick a prime P such that N is not divisible by P. Then pick a random index I (0 Another method is to pick a random key K and sort the entries of the playlist based on HMAC(SONG, K) (where SONG is the name or other identifier of the song). This has a number of interesting properties. For starters the…

I think you're solving the wrong problem here. The goal is not to take a list and shuffle it once. That's trivial. The goal is: given a set of N songs, sample an arbitrarily long sequence of songs with replacement from this set such that the sequence "feels" random. Things that are random but don't "feel random" include: playing the same song too often, not playing a song often enough, and repeating the same sub-sequ…

Also:

- don't play too many songs from the same artist in a row (and too many is probably 2) - don't play too many slow/fast/sad/angry/... songs in a row - ... instruments - ... genres - ... "feel" - ...

Deep AI is definitely needed to create playlists that "feel random". :)

Re: Show HN: A Set of Dice That Follows the Gambler's Fallacy

#189

Post by weaksauce »

bryanlarsen wrote: post #188
rcthompson wrote: post #185

Earlier quoted context omitted.

I think you're solving the wrong problem here. The goal is not to take a list and shuffle it once. That's trivial. The goal is: given a set of N songs, sample an arbitrarily long sequence of songs with replacement from this set such that the sequence "feels" random. Things that are random but don't "feel random" include: playing the same song too often, not playing a song often enough, and repeating the same sub-sequ…

Also: - don't play too many songs from the same artist in a row (and too many is probably 2) - don't play too many slow/fast/sad/angry/... songs in a row - ... instruments - ... genres - ... "feel" - ... Deep AI is definitely needed to create playlists that "feel random". :)

I don't think what people want is random when they say that... they want intelligently mixed playlists that meet a few criteria. no back to back songs from the same artist unless the pool of songs doesn't have sufficient variability in artists, not too frequent repeating of the same song, if you have a live version of the song it should be lowered sufficiently in probability of playing when another cover/live version played. basically people want to remove repetition.
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