Probability and Statistics

 How often is too often?

 From: Gary Wong Address: gary@cs.arizona.edu Date: 2 October 1998 Subject: Re: Netgammon's Dice Forum: rec.games.backgammon Google: wtsoh6dd0k.fsf@brigantine.CS.Arizona.EDU

```Casual_Observer writes:
> If you really want a solid dose of dice
> frustration - go play on the MS Zone for a
> few days. I was fortunate enough this evening
> to be the recipient of 5 rolls in a row that
> were 51. And if you are a fan of multiple
> doublets in a row, this is the place. I have
> had as many as 4 or 5 in a row both in my
> favor and against me. I have even seen, much
> to my wondering eyes, 12 and 14 sets of doublets
> appear in a single game. I grant you, the server
> does have its good moments, but overall, I have
> had games come my way and taken away by
> inordinately ill distributed dice.

I was going to post the probabilities of each of those events happening,
but I guess a bunch of numbers is roughly as interesting to read as the
phone book.  Instead, here's what someone who played a few games every
night with fair dice might typically observe (for themselves or for their
opponent):

2 identical rolls in a row: a couple of times a game
3 identical rolls in a row: once a night
4 identical rolls in a row: a couple of times a month
5 identical rolls in a row: once a year
6 identical rolls in a row: a few times in your life
7 identical rolls in a row: most of us will never see it

(Rolling 51 then 15 counts as 2 identical rolls in a row.)

2 doubles in a row: a couple of times a game
3 doubles in a row: a couple of times a night
4 doubles in a row: a couple of times a week
5 doubles in a row: once or twice a month
6 doubles in a row: a few times a year
7 doubles in a row: once every couple of years
8 doubles in a row: a few times in your life
9 doubles in a row: once in your life
10 doubles in a row: most of us will never see it

(Rolling 44 then 22 counts as 2 doubles as a row.)

The number of doubles rolled in a game depends so strongly on the
distribution of lengths of game that they can't really be estimated
by a simple model (it depends on the players, anyway -- two very pure
and positional players would expect to roll many more doubles per game
than two "run and play safe"rs).

FIBS generates roughly a million dice rolls a week.  This is probably
as many dice rolls as a casual player would make in ten years!  They're
being generated so fast that we would expect several FIBS players a
week to see 5 identical rolls in a row (just like the 5 51's you got
on the Game Zone).  Every couple of weeks, expect somebody to be given 6
identical rolls in a row.  4 doubles in a row happens several times an
hour, and 5 doubles in a row many times a day.

If EVERYBODY who rolled something 5 times in a row (like you did)
posted about it on r.g.b., then by the time you add all the FIBS and
Yahoo and Game Zone and Games Grid and Netgammon players out there
(not to mention those playing against their own computer, and those
playing real games with real dice), then we would be FLOODED with
several posts per day complaining about "inordinately ill distributed
dice"!  Yes, your experience was uncommon (most players would only see
that happen to them about once a year); but there are so many people
out there that I am sure it happens to several people every day.
Since your sample was self-selected, I don't regard it as any evidence
that the Game Zone dice are unfair.

Cheers,
Gary.
--
Gary Wong, Department of Computer Science, University of Arizona
gary@cs.arizona.edu     http://www.cs.arizona.edu/~gary/
```

### Probability and Statistics

Average game and match length  (JP White, Dec 2000)
Average luck of each roll  (Timothy Chow+, Mar 2013)
Average luck of each roll  (Jørn Thyssen+, Feb 2004)
Calculating winning chances  (Douglas Zare, June 2000)
Chance of rolling x doubles in y rolls  (Raccoon+, July 2007)
Chance of rolling x or more pips in y rolls  (Tom Keith, Feb 2004)
Clumping of random numbers  (Gary Wong, Sept 1998)
Counting shots  (Koyunbaba+, June 2007)
Counting shots  (John Little+, Mar 2007)
Distribution of points per game  (Roland Sutter, June 1999)
Distribution of points per game  (Stig Eide+, Sept 1995)
Expected variation in points after a series of games  (Achim Müller+, Feb 1999)
How many games to decide who's better?  (Stephen Turner, Mar 1997)
How often is too often?  (Gary Wong, Oct 1998)
Losing after bearing off 14 checkers  (Daniel Murphy, July 1999)
Number of games per match  (Jason Lee+, Jan 2005)
Number of rolls to enter x checkers from bar  (Michael Depreli+, Mar 2011)
Visualizing odds  (Daithi, Mar 2011)