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13.3  Algebra of events

Operations on events are the same as those on sets. It is not surprising as events are actually sets.

Definition 13.3.1

We say that two events in a certain sample space Ω are identical if they have the same sets of the favorable elementary events.


In the example 13.2.1 events D = "the number of pips is a square of a natural number" and E = "the number of pips is congruent to 1 modulo 3 " are identical, as the very same elementary events are favorable to them. In the rolling two dice trial, see example 13.2.2, the event C = "the product of pips received is an even number" and the event "there is at least one even outcome" are identical events as the very same elementary events belong to both of them.


Definition 13.3.2

Let A, B be events, A, B ⊆ Ω. Then the event A ∪ B is called a union of two events.
The event A ∩ B is called an intersection of two events.
The event A' = Ω - A is called a complement of the event A.


According to the definition of the set theoretical operations, the same elementary events are favorable to the union A ∪ B  which are favorable to either the event A or the event B. In other words, if A and B are events in an experiment with sample space Ω, then the union of A and B is the event that occurs if and only if A occurs or B occurs. Only those elementary events are favorable to the event A ∩ B which are favorable to both the event A and the event B. Only those elementary events of considered space, which are not elements of A, are favorable to the event A' . 

Definition 13.3.3

We say that two events A and B are mutually exclusive if and only if A ∩ B = ∅.


If A and B are disjoint events in an experiment, then they are mutually exclusive. By definition they  cannot both occur on the same run of the experiment.

Example 13.3.1

  1. Let's consider the trial with two dice and the events A, B, C, described in the previous examples, see Figure 13.2.1. Additionally let D = "5 received at least once". The intersection of the events A and B is the event {(1,6), (6,1)}. The intersection of the events A and D, is the event "the sum of pips is 11". There are only 2 events favorable to this event {(5,6), (6,5)}. A complement of the event D is the event "5 never received" . There are 25 elementary events favorable to the event D'.

  2. There is a standard deck of 52 playing-cards. Let us consider a trial of drawing 2 playing-cards one after the other, returning drawn card back to the box (sampling with replacement). Event A = "an ace drawn each time" and B = "10 drawn the second time" are  mutually exclusive. There are no elementary events which are favorable to both events  A and B at the same time. However, the event which is the union of two events A and B has 4*4 + 52*4 favorable events.

Question 13.3.1: What is the power of a set of elementary events favorable to the event "the number of pips received at the first die is greater than the number of pips received at the second die" in a rolling two dice trial?


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