If you cannot afford either the time or the money to take a class on statistics, I recommend Reiner Knizia's "Dice Games Properly Explained" and "Scarne On Dice." Both of these books have excellent chapters discussing dice and how to determine probabilities of outcomes. These are vital books, especially if you wish to become a game designer. I find that the biggest weakness of many games is the designer's lack of understanding of basic statistics (or bad application of them when the designer does have an understanding), or a failure to explain the underlying statistical engine of a game to the players. The

*Dream Park*role playing game by Mike Pondsmith is an example of an otherwise great game that has some serious statistics problems in its basic mechanics, and the otherwise brilliant

*Feng Shui*game does a poor job of stating flat out that the average bonus toward success that a player's given die roll provides is zero. For proper play, games like these either need tweaking or careful adventure design by the game master. This is especially true in

*Feng Shui*where the addition of a "mere" 3 points to a villains attributes/skills can significantly affect probabilities. For example...did you know that in all "balanced" encounters in 4e Dungeons and Dragons, that level is essentially a meaningless construct? Since the players and monsters advance on the same linear path, To hit = x + level, Defense = y + level, Skill DC = x + level...the probabilities set at first level essentially remain true throughout the game. Only the length of combat changes.

Let me be the first to admit that I am no professor of statistics myself. I have taken three courses on statistics, which makes me good enough at them to make stupid mistakes in algorithm designs but hopefully smart enough to admit when I've made a mistake. This is one of the reason I often harass other gamers, who are in fact professors of statistics, to either review my stuff or to help design an analysis. When I do have stuff reviewed, it tends to be very good. When I don't...invariably there is an error. Ugh.

This is because statistics aren't always intuitive. One of the questions that Nobel Prize winner Daniel Kahneman has researched throughout his career is whether we have an "innate" or "subconscious" ability to make probabilistic determinations. In his excellent book

*Thinking, Fast and Slow*, Kahneman gives a nice overview of his life's work on the human mind's ability to count, make correlations, detect patterns, and whether we are good at "intuitive" statistics. What he found, and others, is that the human capacity to recognize patterns and to make associations also makes us very poor at intuitive statistics. Our mind can at a split second -- and without effort -- make all kinds of calculations and recognize associations, but to accurately figure out probabilities takes work. Our very ability to make associations works against the skills needed to apply statistics. Thankfully, we are good at analytical thought -- but that takes more effort than our associative abilities.

One of the key ways that our ability to recognize and induct from patterns, a wonderfully useful skill, is in the "law of averages." There is no such thing. It seems like there should be, but there isn't. If the random events are independent of one another -- meaning that prior acts don't affect future ones -- it doesn't matter how many times you've flipped a coin and had it turn out to be heads. The play

*Rosencrantz and Guildenstern are Dead*plays with this concept wonderfully.

In fact, game designer and podcaster extraordinaire Geoff Engelstein has a nice discussion of this misleading "law" that often infects our minds as gamers.

Note that all of the above applies to the use of dice -- or other independent randomizers -- and not to the use of cards. Cards aren't independent in their randomization. What cards have been used affects what cards remain available -- but that is a discussion for another post.