Raising the bar

Chapter 4. Probability

Unit 4.3. Set versus probabilistic terminology

Random experiment. A random experiment is a process leading to two or more possible outcomes, with uncertainty as to which outcome will occur.

The example of rolling a die is sufficient to illustrate this intuitive definition.

If you stay at the intuitive level, you will never really understand the true meaning of most mathematical constructions.

Table 4.2 helps you establish ways to think about outcomes. When more than one term exists to express the same thing, try to use the more geometric term.

Table 4.2. Set terminology and probabilistic terminology.

Comments.

1. In case of the die the basic outcomes are the sets $latex {1},\,{2},\,{3},\,{4},\,{5},\,{6}$. All other events can be obtained from these 6. For example, the event $latex E$ that an even number turns up equals $latex E = {2,4,6}$. Of course, these numbers cannot turn up at the same time (this is expressed as $latex {2} \cap {4} \cap {6} = \varnothing$). We say that $latex E$ occurs when one of the numbers 2, 4, 6 turns up.
2. We distinguish a number 2 from a set $latex {2}$. Arithmetic operations apply to numbers, while for sets we use set operations.

Disjoint coverings. In some problems we need disjoint coverings. For motivation, think about a jigsaw puzzle. Its pieces possess two properties: any two pieces are disjoint and together the pieces cover (comprise) the whole image. Departing from this geometric example, we say that events $latex {A_1},...,{A_n} \subset S$ are mutually exclusive if any pair $latex {A_i},{A_j}$ is disjoint when $latex i \ne j$ and collectively exhaustive if $latex S = {A_1} \cup ... \cup {A_n}$.

Disjoint coverings, by definition, are sets of events that are mutually exclusive and collectively exhaustive.

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