Basic concepts of probability theory

Hereinafter, instead of speaking «the set of conditions S holds» we shall speak briefly: «the trial has been made». Thus, an event will be considered as a result of a trial.

Example. A shooter shoots in a target subdivided into four areas. One shot is the trial. Hit in a certain area of the target is an event.

Example. There are colour balls in an urn. One takes at random one ball from the urn. An extracting a ball from the urn is the trial. An appearance of a ball of a certain colour is an event.

Events are incompatible if an appearance of one of them excludes an appearance of other events in the same trial. Otherwise, they are compatible.

Example. A detail is extracted at random from a box with details. An appearance of a standard detail excludes an appearance of a non-standard detail. The events «a standard detail has appeared» and «a non-standard detail has appeared» are incompatible.

Example. A coin is tossed. An appearance of «heads» excludes an appearance of «tails». The events «heads have appeared» («the coin lands on heads») and «tails have appeared» («the coin lands on tails») are incompatible.

For example, a landing two different prizes under only one ticket of a lottery are incompatible events, and a landing the same prizes under two tickets are compatible events. Obtaining marks «excellent», «good» and «satisfactory» by a student at an exam in one discipline are incompatible events and an obtaining the same marks at exams in three disciplines are compatible events.

Some events form a complete group if in result of a trial at least one of them will appear. In other words, the appearance of at least one of events of a complete group is a reliable event.

In particular, if events forming a complete group are pairwise incompatible then in result of a trial one and only one of these events will appear.

Example. Two tickets of a money-thing lottery have been bought. One necessarily will happen one and only one from the following events: «a landing a prize on the first ticket and a non-landing a prize on the second», «a landing a prize on both tickets», «a non-landing a prize on the first ticket and a landing a prize on the second», «a non-landing a prize on both tickets». These events form a complete group of pairwise incompatible events.

Example. A shooter has made one shot in a target. One necessarily will happen one from the following two events: hit, miss. These two incompatible events form a complete group.

Events are equally possible if there is reason to consider that none of them is more possible (probable) than other.

Example. An appearance of heads and an appearance of tails at tossing a coin are equally possible events. Appearances of «one», «two», «three», «four», «five» or «six» on a tossed die are equally possible events.

Several events are uniquely possible if at least one of them will necessarily happen as a result of a trial. For example, the events consisting in that a family with two children has: A – «two boys», B – «one boy and one girl» and C – «two girls» are uniquely possible.