In the previous column, we considered some subtleties of throwing multiple-sided dice. But there are even more interesting nuances to consider. In the normal way of throwing dice, the results for each die are considered to be independent events. In many normal dice games, 1, 2, or 3 dice are thrown. When multiple dice are thrown, their values are combined in a way that assumes no connection between them.

However, in several past articles I have tried to alert readers to the way in which subtle changes in the wording of a problem or the conditions of a measurement can radically change the probability of the outcome. The classic example of this is the frustratingly persistent arguments that the Monty Hall paradox generates. (Reminder: Monty allows a contestant to choose one of three doors to open. The contestant gets what is behind the door. Two doors have trash. The other has a treasure. After the contestant chooses and before he looks, Monty opens one of the unchosen doors and shows that it has trash. He then asks the contestant if he wants to keep his first choice or switch to the other unopened door. Many people think it makes no difference, but in fact, the odds of winning are increased to 2 to 1 by switching. If you don’t believe this, see my tutorial or search for the paradox explanation on the Internet.)

In the example where I proposed one way to generate random numbers from 0-39 with two dice numbered from 0-19, a non-random little coupling of the independent die was proposed. The uniform distribution is generated by making the die distinguishable in some way and designating one of them to generate the “tens” column and the other the “ones” column of the desired range of random numbers. So far so good, but suppose that the thrower is allowed to choose which die is “tens” and which is “ones” after they have been thrown and the results known. This option obviously adds a non-random factor since the player would presumably make a choice that favors winning or getting extra points or whatever.

Such coupling of independent events is common in games of all types. For dice throws, the plays one can make in backgammon depend on the combination of independently thrown dice. Cards are dealt at random in poker, but the process of throwing down selected losers and drawing replacements introduces a non-random distribution. The skill in doing that coupled with other skills – such as reading the other players – makes the difference between who ends up winning and losing.

In bridge, the hands are dealt randomly, but the process of bidding partially destroys that randomness by giving information from one random hand to another. By the time the play starts, experts will have a good feeling for what the other players are holding even though the cards have not been shown.

Once you get into the spirit of looking for the ways in which pure randomness is broken by coupling events, it becomes a fun game.

Maybe in a future column we can show how this concept applies to picking stocks – not that I have a system or advocate one!

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.