Probability and statistics are related areas of mathematics which concern themselves with analyzing the relative frequency of events. Still, there are fundamental differences in the way they see the world:

ˇ    Probability deals with predicting the likelihood of future events, while statistics involves the analysis of the frequency of past events.   

• Probability is primarily a theoretical branch of mathematics, which studies the consequences of mathematical definitions. Statistics is primarily an applied branch of mathematics, which tries to make sense of observations in the real world.

Both subjects are important, relevant, and useful. But they are different, and understanding the distinction is crucial in properly interpreting the relevance of mathematical evidence. Many a gambler has gone to a cold and lonely grave for failing to make the proper distinction between probability and statistics.

This distinction will perhaps become clearer if we trace the thought process of a mathematician encountering her first craps game:

ˇ      If this mathematician were a probabilist, she would see the dice and think ``Six-sided dice? Presumably each face of the dice is equally likely to land face up. Now assuming that each face comes up with probability 1/6, I can figure out what my chances of crapping out are.''

• If instead a statistician wandered by, she would see the dice and think ``Those dice may look OK, but how do I know that they are not loaded? I'll watch a while, and keep track of how often each number comes up. Then I can decide if my observations are consistent with the assumption of equal-probability faces. Once I'm confident enough that the dice are fair, I'll call a probabilist to tell me how to play.''

In summary, probability theory enables us to find the consequences of a given ideal world, while statistical theory enables us to to measure the extent to which our world is ideal.

Modern probability theory emerged from the dice tables of France in 1654. Chevalier de Méré, a French nobleman, wondered whether the player or the house had the advantage in a variation of a particular betting game. In the basic version of this game, the player rolls four dice, and wins provided none of them are a six. The house collects on the even money bet if at least one six appears.  

De Méré brought this problem to attention of the French mathematicians Blaise Pascal and Pierre de Fermat, most famous as the source of Fermat's Last Theorem. Together, these men worked out the basics of probability theory, along the way establishing that the house wins the basic version with probability    ,     where the probability p = 0.5 would denote a fair game where the house wins exactly half the time.    The jai-alai world of our Monte Carlo simulation assumes that we decide the outcome of a point between two teams by flipping a suitably biased coin. If this world were reality, our simulation will compute the correct probability of each possible betting outcome. But all players are not created equal, of course. By doing a statistical study of the outcome of all the matches involving a particular player, we can determine an appropriate amount to bias the coin.