One of my favorite parts of mathematics, unsurprisingly perhaps, is probability. Though Einstein once famously said, “God does not play dice,” there indeed do appear to be areas where chance is given a great deal of sway in the universe. It is the purpose of this entry to provide some brief brushstrokes to show the outlines of some of the most intriguing aspects of probability.
Some aspects of probability appear hard-wired into the universe. For example, there are aspects of statistical physics that are only comprehensible in terms of statistical distributions rather than any kind of visible and discrete phenomena at all. Other events such as catastrophes and tragedies appear to be, in part, “time and chance” effects that afflict the righteous and the unrighteous alike. Not all disasters and tragedies are judgment for sins–some are merely the result of being the wrong place at the wrong time.
Other aspects of time and chance serve to tantalize and frustrate scientists and philosophers. For example, probabilities are often far trickier than we would like to imagine. The calculation of probabilities is something that does not come easy, as we tend to magnify the probabilities of that which we wish to happen and minimize the probabilities of what which we do not wish to happen. Two examples of this are most interesting in terms of worldview. Blaise Pascal’s famous “wager” about the “lottery ticket” of belief in God was turned around improperly by Laplace to show that the reliability of someone providing revealed information of God was extremely unlikely the higher the reward promised. Unfortunately Laplace never found out the truth of the anthropic principle and never discovered just how “lucky” mankind is to even exist, odds so vanishingly small in the odds of scientific constants and other physical properties that the likelihood of intelligent design is very nearly at unity. Inversely, the odds of unguided Darwinian evolution are so vanishingly small that they appear to be beyond the universal probability boundary of 1 x 10^-150, one of the factors influencing many scientists to support the multiverse conjecture in order to inflate the probabilistic resources of the universe.
Still other aspects of probability excite the common person. Card games and dice games, or gambling on sports from horse racing to football to an NCAA Bracket Pool, is essentially an exercise in probability. People tend to believe that their success is due to skill (and if one is skilled at card counting, certain games like Blackjack can be won by skill) and that their failure is due to luck, if they are confident (often mistakenly so) about their abilities. Even love is to some extent an exercise in probability. Let us assume that there is a given proportion of suitable future husbands or wives in a given population (depending on one’s qualifications). Success would require a mutually recognized correspondence of suitability, both people knowing each other when both were free to find attachments, and enough chemistry and attraction to pursue a relationship and enough persistence to overcome obstacles along the way. If someone’s requirements are sufficiently demanding, the odds of success become vanishingly small, regardless of how well they take advantage of their opportunities. At this moment one hopes that one ceases to deal with random chance and probability and is dealing with divine providence working the unlikely odds in one’s favor.
As probability is such an important aspect of game theory as well as understanding important and puzzling aspects of the world in which we live, it is something I have long studied. Nonetheless, probability quickly moves into dangerous and unexpected territory as soon as one leaves the confines of dice and coin flips and enters into the messy and complicated world of human behavior and historical events. That said, perhaps the messiness of probability is precisely what makes it most appealing to me. Perhaps.