3.3 Venn diagrams and the algebra of events
A graphical depiction of occasions that is very valuable for showing logical connections among them is the Venn diagram. The sample space S is stood for as consisting of all the points in a huge rectangle, and also the events E,F,G,…, are stood for as consisting of all the points in offered circles within the rectangle. Events of interest have the right to then be indicated by shading proper areas of the diagram. For instance, in the three Venn diagrams shown in Figure 3.1, the shaded locations reexisting, respectively, the occasions E∪F, EF, and also Ec. The Venn diagram of Figure 3.2 suggests that E⊂F.
You are watching: The intersection of two events a and b is the event that:
Figure 3.1. Venn diagrams. (A) Shaded region: E ∪ F; (B) shaded region: EF; (C) shaded region: Ec.
Figure 3.2. Venn diagram.
The operations of developing unions, intersections, and also complements of events obey certain rules not dissimilar to the rules of algebra. We list a couple of of these.
|Commutative law||E ∪ F = F ∪ E||EF =FE|
|Associative law||(E ∪ F)∪G = E ∪ (F ∪ G)||(EF)G = E(FG)|
|Distributive law||(E∪F)G=EG∪FG||EF ∪ G = (E ∪ G)(F ∪ G)|
These relationships are confirmed by reflecting that any type of outcome that is included in the event on the left side of the ehigh quality is also consisted of in the occasion on the appropriate side and also vice versa. One method of showing this is by implies of Venn diagrams. For circumstances, the distributive legislation might be confirmed by the sequence of diagrams shown in Figure 3.3.
Figure 3.3. Proving the distributive regulation. (A) Shaded region: EG; (B) shaded region: FG; (C) shaded region: (E ∪ F)G, (E ∪ F)G = EG ∪ FG.
The adhering to beneficial connection in between the 3 standard operations of developing unions, intersections, and also complements of occasions is well-known as DeMorgan's laws.
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Sheldon M. Ross, in Introductory Statistics (4th Edition), 2017
Experiment: Any process that produces an observation.
Outcome: The monitoring developed by an experiment.
Sample space: The set of all possible outcomes of an experiment.
Event: Any collection of outcomes of the experiment. An occasion is a subset of sample area S. The occasion is sassist to happen if the outcome of the experiment is contained in it.
Union of events: The union of events A and also B, denoted by A∪B, is composed of all outcomes that are in A or in B or in both A and also B.
Intersection of events: The interarea of occasions A and B, dedetailed by A∩B, is composed of all outcomes that are in both A and also B.
Complement of an event: The complement of event A, dedetailed by Ac, consists of all outcomes that are not in A.
Mutually exclusive or disjoint: Events are mutually exclusive or disjoint if they cannot take place at the same time.
Null event: The event containing no outcomes. It is the complement of sample area S.
Venn diagram: A graphical representation of events.
Probability of an event: The probcapacity of event A, deprovided by P(A), is the probcapability that the outcome of the experiment is contained in A.
Addition dominance of probability: The formula
Conditional probability: The probcapacity of one occasion given the indevelopment that a 2nd occasion has occurred. We denote the conditional probcapacity of B given that A has actually occurred by P(B|A).
Sheldon M. Ross, in Introductory Statistics (Third Edition), 2010
Any set of outcomes of the experiment is dubbed an occasion. We designate occasions by the letters A, B, C, and so on. We say that the event A occurs whenever before the outcome is consisted of in A.
For any kind of 2 occasions A and also B, we define the new occasion A ∪ B, called the union of events A and B, to consist of all outcomes that are in A or in B or in both A and also B. That is, the occasion A ∪ B will occur if either A or B occurs.
In Example 4.1(a), if A = g is the occasion that the boy is a girl and also B = b is the event that it is a boy, then A ∪ B = g, b. That is, A ∪ B is the entirety sample space S.
In Example 4.1(c), let
A = all outcomes starting with 4
be the event that the number 4 equine wins; and also let
B = all outcomes whose second element is 2
be the event that the number 2 horse comes in second. Then A ∩ B is the occasion that either the number 4 equine wins or the number 2 steed comes in second or both.
A graphical representation of occasions that is extremely useful is the Venn diagram.
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The sample space S is represented as consisting of all the points in a large rectangle, and occasions are represented as consisting of all the points in circles within the rectangle. Events of interemainder are shown by shading proper areas of the diagram. The colored area of Fig. 4.1 represents the union of events A and B.