Repeated die roll - is it truly random ?

[danielvictor]

Ok,I had been noticing this - then it seemed to go away,but maybe it happened again just now. I rolled the same number on a d-8 three times in succession.Ok,I get that I''m bound to roll it twice one time in 8,so then it's only not-random if the third die-roll repeats again more than 1 time in eight after the second did. Only it seems to do that Has anyone else noticed that ? [no,I don't have time to clear cache and cookies between die-rolls !].

[Gulchen]

Consier the Markor process
https://en.wikipedia.org/wiki/Markov_chain
with states

3: ​ at least three times in succession occurred
2: ​ 3-in-a-row did not occur, and the current streak has length 2
1: ​ 3-in-a-row did not occur, and the current streak has length 1

.


I wrote Julia code to compute the probabilities for each of the above states for different numbers of rolls:
prevprobs holds the probabilities for the previous number of rolls, and nextprobs holds the probabilities for the previous number of rolls.



Version 1.11.4 (2025-03-10)

Code: Select all

julia> z = (BigInt(0))//(BigInt(1))
0//1

julia> typeof(z)
Rational{BigInt}

julia> iszero(z)
true

julia> prevprobs,nextprobs = [z,z,z],[z,z,z]
(Rational{BigInt}[0, 0, 0], Rational{BigInt}[0, 0, 0])

julia> nextprobs[1] = (BigInt(1))//(BigInt(1))    #  The first roll gives a streak of length 1.
1//1

julia> "Now, nextprobs holds the probabilities for just after 1 roll.";

julia> for i in 1:3
               println(i)
       end
1
2
3

julia> "The following loop runs exactly 50 times,";

julia> for i in 1:50
               prevprobs[1],prevprobs[2],prevprobs[3] = nextprobs[1],nextprobs[2],nextprobs[3]    #  Move values from nextprobs to prevprobs,
               nextprobs[1],nextprobs[2],nextprobs[3] = z,z,z    #  and then clear out those values from nextprobs.
               nextprobs[2] += (prevprobs[1])*((BigInt(1))//(BigInt(8)))    #  There is a 1/8 chance of extending the streak from length 1 to length 2,
               nextprobs[1] += (prevprobs[1])*((BigInt(7))//(BigInt(8)))    #  and a 7/8 chance of getting a new streak, which will have length 1.
               nextprobs[3] += (prevprobs[2])*((BigInt(1))//(BigInt(8)))    #  There is a 1/8 chance of extending the streak from length 2 to length 3,
               nextprobs[1] += (prevprobs[2])*((BigInt(7))//(BigInt(8)))    #  and a 7/8 chance of getting a new streak, which will have length 1.
               nextprobs[3] += prevprobs[3]    #  If it was the case that some 3-in-a-row occurred, then it's still the case that some 3-in-a-row occurred.
       end

julia> "so now nextprobs holds the probabilities for just after 1+50 rolls, and prevprobs holds the probabilities for just after 50 rolls.";

julia> prevprobs[3] < (BigInt(42))//(BigInt(85))
true

julia> (BigInt(215))//(BigInt(429)) < nextprobs[3]
true

julia> nextprobs[3] < (BigInt(214))//(BigInt(427))
true

julia> 

Thus, 51 rolls gives probability strictly between 215/429 and 214/427 of getting the same number on a d-8 three times in succession.

( ​ 215/429 and 214/427 are each slightly greater than 1/2. ​ )

[danielvictor]

That's interesting,but what concerns me is whather after you roll the same number twice in succession,the third die roll is still random and only repeats one time in eight.My personal experience has been that it repeats more often than that. I have to disregard the second repetition,because it will repeat one time in eight,and I may only notice when it does repeat.The fact that nothing has altered other than the number of die rolls left also makes it appear subjectively as if it hasn't worked properly.

[Silene]

My personal experience has been that it repeats more often than that.

I guess this means that you have counted. So please present your numbers and how you achieved them. So we can check if they are within statistical boundaries or not.

[Gulchen]

@ danielvictor ​ :


Is this for

specifically your rolls
or
each player's rolls in your games, counted separately for each player
or
all rolls in your games, no matter which player does each roll

?

[danielvictor]

It is specifically for my rolls,and I haven't kept enough detailed notes to present it as a bug.That is why I was asking others whether they had observed the same thing on this forum.

[Silene]

It is specifically for my rolls,and I haven't kept enough detailed notes to present it as a bug.That is why I was asking others whether they had observed the same thing on this forum.

Ah ok. So just a gut feeling.

Please do not forget that gut feelings are often wrong. Our brain is incapable to process all information at the same intensity so it makes us believe to see patterns.

But apart from that, it is normal in randomness that exceptional results happen in small sample sizes. So every now and then, everyone can observe some kind of pattern for a while. It'll pass.

[danielvictor]

It's more than just a gut feeling. It's a memory of what happened,even though that memory may be subjectively biased. I think that that has been happening a lot more than one time in eight,but I cannot be certain.
When I thought it was happening,I began re-starting my computer between die rolls,and switching browsers. Then the die rolls stopped repeating so I stopped bothering,but it happened again so I wrote that post. I play on a PC,using Google Chrome.

[Jellby]

I'm pretty sure the rolls are done on the server, so anything you do on your computer is irrelevant.

Start taking notes now of every roll, and when you have enough data you can evaluate your memory.

[Gulchen]

@ danielvictor ​ :



Your 17th-most-recent game had at least two d8 rolls by you.
I looked at [the last two d8 rolls by you in that game] and
[all the d8 rolls by you in your sixteen most recent games].

The last two in your 17th-most recent were ​ 2,7 , ​ your 16th-most-recent
had more than two by you, and the first two in that game were ​ 4,5 ,
so allowing crossover from your 17th-most recent game doesn't matter.

You had exactly four consecutive 1s in your 16th-most-recent game,
and those were not the last four rolls of that game.
This gives [2 times that two-in-a-row was followed by
another of the same value] and [1 time that two-in-a-row
was followed by a roll with a different value].


Other than those 3 times:

There was exactly 10 sequences of three consecutive
d8 rolls by you _within the same game_ where the
first two of those rolls gave the same value.
If one drops the ​ within the same game ​ part,
one gets exactly 3 more such sequences.
Each of your 16 most recent games had at least one d8 roll
by you, and each of those ​ 3 more such sequences ​ used
only two consecutive games, not three consecutive games.
For each of the 10 sequences where the rolls were within a single game,
_and_ each of the 3 more without the ​ within the same game
part, the last roll in the sequence gave a different value.


Thus, depending on how one counts, it was either

2 times out of 13
or
2 times out of 16

that after you rolled the same number twice in succession
with a d8, the third d8 roll gave the same value.