(This document is held on the server so that we can update and improve our explanation over time, without needing new releases of the app.)

Armed with our stats tables the player is invited to interpret these for the purposes of assessing fairness. However this needs some background.

First consider the most common type of example given to claim that the dice are unfair:

"My opponent just threw a double 6 three times in succession! The chance of that is impossible! It is (1/36) x (1/36) x (1/36) or about 0.00214% chance! or 1 in 46656! The program absolutely certainly cheats!!"

That would be true if you sat down and then immediately threw 2 dice 3 times and saw 3x double 6. In that case the probability above is correct, and indeed extremely unlikely!

However that is probably not what happened. Most likely the player will have been playing games and then this suddenly happened. The player was given many chances to see this suspect throw. In just a single round of Backgammon each player will have had on average 25 throws. Within that round there will have been 25-2 = 23 opportunities to throw this 3x double 6. Given that this event may have happened after 20 rounds (which might be completed inside an hour), then we can calculate the probability that 3x double 6 could have been thrown in 20 rounds.

This makes the calculation a little more complex as, of course, in 20 rounds we might have thrown any number of 3x double 6 and there will be a probability that we threw 1, another probability we could throw 2 etc. However to answer our question of "what is the probability that we threw any 3x double 6s at all" we would need to add these all together. That is too complex, so instead we can calculate the opposite simpler question of "What is the chance of not throwing a 3x double 6?

To calculate the probability of not throwing a 3x double 6 in 20 rounds we first need to subtract the chance of throwing it once (0.00214%) from 100%, which gives us 99.99786%, which is the chance in a single attempt that we will not throw 3x double 6. As a probability fraction, this is 0.9999786. In 20 average rounds of Backgammon, there will have been 23 x 20 = 460 chances to throw 3x double 6. The chance that none of these opportunities resulted in 3x double 6 is calculated as:

(0.9999786 x 0.9999786 x 0.9999786 x…. 460 times), which is 0.9902

This would normally be expressed by this formula, 0.9999786 raised to the power 460:

This means that there is a 99.02% chance that this throw was not made, and therefore a 100% - 99.02% = 0.98% chance of throwing 3x double 6 with fair dice. While this is still unlikely, it is clearly possible.

However, we also have to consider that many people all over the world are playing this game. When we last checked there were about 120,000 players per day. If we consider only 1% of these players (1,200 players), we can calculate the chance that none of these players would have seen 3x double 6:

This means that the chance that none of these 1,200 players would have seen 3x double 6 is only 0.000737%. In other words, the odds are 1 in 135,673 that none of these players would have seen 3x double 6 with fair dice. Therefore, it is very likely that many players would have seen this event.

However with 120,000 players playing at once, not 1,200, then the above calculation becomes:

Given that there are 120,000 players per day, it is certain that many hundreds of these players will see the app throw 3x double 6. While some of these players may suspect that the program is cheating, this is not proof of cheating. Fair dice will result in many hundreds of players seeing this throw.

What about the stats in the tables?

One way to determine if the distribution of doubles is fair is to use a binomial distribution. For example, if you have seen 60 doubles while the CPU has had 90 doubles, this may seem unfair. However, by using a binomial calculation, you can determine the probability that the player is getting an even distribution of doubles in the long term. In this case, the probability would be 0.7180%, or 1 in 139.

Surely that means it is cheating?!?

No. In the case above, there are 120,000 players playing each day, and all of them have a chance of seeing a skewed result even with a fair distribution of doubles. While the math behind this may not be immediately obvious, it is possible for a fair distribution to produce such a skewed result as 60 doubles for the player and 90 doubles for the CPU.

This can be demonstrated through a simulated run, as shown in www.aifactory.co.uk/backgammon.htm. The simulation shows the number of doubles for the player and CPU over 20 rounds, with an equal chance for each to throw a double, repeated for 120,000 simulated players. On average, each simulated player had 75 doubles over 20 rounds, but 688 of these players saw a double rate of only 40% or less (60 or fewer doubles compared to the CPU's 90 or more).

This means that even with a fair distribution of doubles and 120,000 players playing simultaneously, the simulation showed that about 688 players would see themselves getting 60 or fewer doubles while the CPU had 90 or more. So if you see this level of skew in your own game, it is important to realise that with fair dice you may just be one of the 688 players who will have inevitably seen such a skewed result.