Introduction to Probability Theory

We are surrounded by numbers. They can conceal the truth. Numbers can impress. There are even numbers that will cost you big bucks. Your odds of buying a winning lotto ticket are 1 in 35; 1 in 5,245,786 if you want to win the grand prize. There are 221,184 ways to customize your Burger King Whopper. And if you are twenty-five, odds are that you will live for another 53.5 years.

We are surrounded by numbers. Often, we have no idea how the numbers were calculated or what they really mean. One type of numbers that many people do not understand are generated by probability theory.

Probability theory got its start as a science in the seventeenth century. While there are examples of some of its basic techniques being used before that time, it was not until the French mathematicians Blaise Pascal and Pierre de Fermat started to discussing the underlying behavior of random games of chance that it became a branch of mathematics. Today, probability theory is used not only by gamblers; it is used by physicists, meteorologists, drug companies, and government agencies among others.

The basic idea of probability theory is that by figuring out all the outcomes from a set of actions, the probable outcomes of future events and their likelihood of occurrence can be predicted. The classic examples are coin-flipping, card and dice games. A more modern example is the prediction of how easily your password can be guessed without resorting to social engineering.

As an illustration, consider the number of ways that you can arrange three objects in a series. For the first slot of the series, you have three choices. This leaves you with a choice of two options for the second slot of the series, and only one for the last slot. Therefore, there are six possible ways to arrange three objects in a series. If you were restricted to three specific letters and only allowed to use them once each in a password, there is a one in six chance that someone else could guess the sequence you arranged them in on the very first try.

The security of computer passwords from mere random guessing rests on how rapidly the numbers of possible arrangements increase. Four letters produces twenty-four possible arrangements; five produces one hundred and twenty. Given doubt about the letters used and the possibility of using passwords of varying lengths, the sheer number of possible passwords makes it unlikely that a person would be able to guess a nonsensical password, especially if they were only given a limited number of guesses. This is why cyber-criminals resort to phishing and data-mining when they are in search of passwords to exploit.

The same method of calculation used in our password example is how the odds of correctly guessing the six lotto numbers are figured. In the case of the Colorado Lottery lotto game, it is 42 x 41 x 40 x 39 x 38 x 37 or 5,245,786 possible arrangements; each arrangement being one possible outcome. Each of these possible outcomes has an equal chance of occurring.

A common misconception about probability theory is that future outcomes are affected by past events; this is called the gambler's fallacy. An example of this misconception is the belief that a coin being flipped is more likely to come up tails after a run of heads. The truth is that on every flip of a coin, it is just as likely to come up heads as it is tails. This belief is what causes people to study numbers previously drawn in the lottery to try to predict the results of future drawings.

Previous experience is not always useless. For instance, when determining life insurance rates, insurance companies use actuarial tables to determine how much longer a person is likely to live. These tables are based on previous data about death rates among various risk groups. This is an example of possibility theory being used in the business world.

There are many other applications of probability theory. Those interested in studying probability theory should prepare by taking as many math courses as possible, including calculus.