The Sample Space, Events and Probabilities

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Probability theory is concerned with the outcomes of random experiments. These can be either real world processes or Gedankenexperimente. In both cases,

  • the experiment has to be infinitely repeatable and
  • there has to be a well-defined set of outcomes.

The set of all possible outcomes of an exeriment is called the sample space which we will denote by S. Consider the process of rolling a die. The set of possible outcomes is the set S=\{1,2,3,4,5,6\}   Each element of \ S is a basic outcome. However, one might be interested in whether the number thrown is even, whether it is a three or a seven, and so on.  Thus we need to be able to speak of various combinations of basic outcomes, that is subsets of S. An  event is defined to be a subset of the set of possible outcomes S. We will denote an event using the symbol E.  Events which consist of only one element, such as ’a two was thrown’ are called simple events or elementary events. Simple events are by definition not divisible into more basic events, as each of them includes one and only one possible outcome. Example:Rolling a single die once results in the occurrence of one of the simple events {1}, {2}, {3}, {4}, {5}, {6}.As we have indicated, the sample space S is {1,2,3,4,5,6}. Example:Tossing a coin twice.Sample space: S=\{TT,TH,HT,HH\}.Simple events: {TT},{TH},{HT },{HH}, T\equivTail , H\equivHead.This specification also holds if two coins are tossed once. It will be convenient to be able to combine events in various ways, in order that we can make statements such as ”one of these two events happened” or ” both events occured”.  For example, one might want to say that ”either a 2 or 4 was thrown”. or ”an even number larger than 3 was thrown.  Since events are sets, (in particular subsets of the set S), we may draw upon the conventional tools of set theory.

Venn diagram

A common graphical representation of events as subsets of the sample space is the Venn diagram.  It can be used to visualise various combinations of events such as intersections and unions.

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