Most everyone has seen the various Pirates of the Caribbean movies and knows of the pirates' dice game commonly played on the Flying Dutchman, Liars Dice.
The players use six dice shaken in cups apiece. They can only look at their dice hidden under their cups but must guess at a given ratio of x dice of y value on the board. As an equation it can look pretty messy, guessing variable n of N of x for y given y is the desired number on the dice.
Let's look at this another way.
The minimum bet is 2,2's where there are at least 2 dice with a value of 2 showing on the table(including wilds which are 1).
The next play can increase one value or both. Let's say 3,4's. The next bet would have to be either 3,5's or 4,4's. The bet number never goes down. IF you are really confident you can start off the bet a bit high, like 5,5's or 5,6's and forcing an early call of dice by declaring the better a liar. Remember that for each player playing another 6 is added to the total.
So how do we cheat the probability im sure you're asking by now.
It's easy. Observe the highest likely possible outcome on the field based on your dice first.
Here's where the quantum inference comes in. The state of an unknown value isn't known until it is observed, which means that observing it makes the value static and known.
So we roll our dice in the cup and get the below values
1, 2, 4, 4, 3, 1
if there are 4 players than theoretically this represents a possible 25% of the values on the board, ranging from 17(13+ minimum value of wilds) to 25(13+maximum value of wilds). This leaves a lot for consideration but the other players do some of that consideration for us.
We will open with a call of 3,2's as this is a combination of possible dice within our roll. The next calls are 3,3's. 4,3's. 4,4's. We know know that 2 of the other 3 players have at least the possibility of 6-8 fours between them given their calls. It is unlikely that a dummy call has been made so early in the game so we shall take them as an accurate assesment. A certain amount of this can also be made up of 1's or wilds, and like with our own selection of numbers we can inflate and deflate true static values with.
Our bet will have to open with a minimum of 4,4's so we can inflate it a bit with half of our own value, making it 6,4's. This is a bold statement and might draw some attention, but if you are called a liar you are likely right and have eliminated a die from the other player as a penalty for lying(or lost some monetary value depending on betting parameters). Safely the bounds could be pushed up to about half of the total if you have another 1 or 4.
Automated and electronically randomized means of creating a dispertion of variables are not influenced by this in such an easily measurable way. The same theory could also be applied to cards if you know what ratios are needed to determine a certain amount of success. You could also use the same to determine in games like poker and black jack where your position of likely expression within a given strata of likelyhood your hand is; and possible permutations of expression and their associated possibility of hapening.
By using our own sample we can infer what we have as a likelyhood of expression of dice, and then use our choice of bets to determine the highest likely number. Once we know our highest likely number we have to test player's organic reactions for margins of error. After a point this becomes easier to figure out and you start playing in the lower range of the upper quartile of tollerable limits as opposed to trying to figure out the highest likely arrangement. Even with a lower 25% range the safety margin is more than acceptable. Remember to take your sample size assement of your own dice after the first round of bets. If we had stuck with 2's for example, the maximum tolerance range would have been 8-12 2's total with the mean of distribution being somewhere around 3 per player. Guesses at higher ranges would have to be inferred from other players.