Understanding important concepts in probability

First of all, we need to clarify some fundamental concepts in probability theory.

Events and sample space

The easiest and most intuitive way to understand probability is probably through the idea of counting. When tossing a fair coin, the probability of getting a heads is one half. You count two possible results and associate the probability of one half with each of them. And the sum of all the associated non-overlapping events, not including having a coin standing on its edge, must be unity.

Generally, probability is associated with events within a sample space, S. In the coin tossing example, tossing the coin is considered a random experiment; it has two possible outcomes, and the collection of all outcomes is the sample space. The outcome of having a heads/tails is an event.

Note that an event is not necessarily single-outcome, for example, tossing a dice and defining an event as having a result larger than 4. The event...