Probability

"Probability" is the part of mathematics which looks for laws governing random events. More specifically, consider some process which can have a variety of different outcomes. Suppose now that, for some reason, we favor one or several of these possible outcomes. The subject of probability addresses the question as to how likely it is that our favored outcome(s) actually arise.

Probability is now a serious and well established part of mathematics. Perhaps not even surprisingly, it has its origins in games of chance, more commonly called gambling. So what does gambling have do with mathematics? - The key is quite simply that all gamblers want to know:

What are my chances of winning?

Due to the insight of Blaise Pascal (1623-1662) we can answer this question in many cases with a mathematical formula. We present this formula below. For now let us just explore its ingredients. In a game of chance, we need to

  1. determine the total number N of possible outcomes of a given chance situation
  2. determine the number W of winning outcomes of the chance situation
  3. if W is greater than N/2, your chances of winning are higher than your chances of losing.

Let us illustrate this method by considering some examples.

A precise measurement of the favorable outcomes   can be obtained by forming the quotient

   P :=  
number of favorable outcomes of the gamble
number of all possible outcomes of the gamble

We call this number P the probability for the event to occur. The number P lies always between 0 and 1. If P is greater then 1/2, chances for the event to occur are higher than the chances for it not to occur. If P is less than 1/2, the chances that the event to occur are lower than the chances for it not to occur.

Here is a more serious application of what we have learnt: Can you fake true randomness? (Be patient! Web browsers need a long time to start the applet in this page.)