Probability - the Australian Mathematical Sciences Institute

{10} • ProbabilityEvents may be expressed in words, inside quotation marks:• In the die-rolling example, we can consider the event A = “a prime number is obtained”= {2,3,5}.• In the example of five people born in 1995, we can consider the event B = “exactly twoof the individuals experience at least one broken leg prior to turning 20”.When expressing an event in words like this, the description must determine a subsetof E . We can regard something as an event only when it is true that, for every possibleoutcome, we can say definitively whether or not the event has occurred.An outcome is sometimes called an elementary event.The following are events in any context:• The event space E is itself an event, because it is a collection of outcomes. We canthink of E as the event “the random procedure occurs”.• The empty set ∅ is a subset of E . The event ∅ is not very interesting; it contains nooutcomes.The mathematical approach to probability requires that we have to be able to specify thepossible outcomes. Outcomes that we may be able to imagine, such as the die comingto rest on a corner (leading to no number being uppermost), are often excluded fromconsideration altogether. If we do want to consider an outcome as possible, it must beincluded in the event space.Recall that the domain of probability statements consists of events. Now that we understandthat events are subsets of the event space E , we see that probability is a setfunction. It maps subsets of E into the interval [0,1].û û ûû û û û ûû û ûû û A ûû ûû ûûûû ûûû û û û û û û û ûûûεPr0 1Probability as a set function.Operations on events and relations between eventsEvents in random procedures are sets of possible outcomes. We now apply ideas aboutsets to events; this will help us to think clearly about the events and, subsequently, theirprobabilities.