User:Godfredabledu/Probability/Syllabus/Background to Probability

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Background to Probability

Historically, probability originated from the study of games of chance and early applications of the theory of probability were in such games. In the middle of the 17th century, a French coutier, the Chavelier de Mere wanted to know how to adjust the stakes in gambling so that in the long run, the advantage would be his. He presented the problem to Blaise Pascal. It was in the correspondence between Pascal and Pierre Fermat, another French mathematician, that the theory of probability had its beginning. Many of the probability calculations were based on objects of gambling: the coin, the die, and cards.

In order to understand probability, it is useful to have some familiarity with sets and operations involving sets. This is because in probability theory, we make use of the idea of set and operations involving sets. Set: A set is a collection of elements. The elements of a set may be people, horses, desks, files in a cabinet, or even numbers.

Universal Set: It is a set containing everything in a given context. We denote the universal set by S.

Intersection of Sets: If sets A and B have elements in common, we say they intersect. The intersection of A and B is denoted AB.

Example: Given that A = {2, 3, 4, 5, 6} and B = {2, 4, 5., 7, 8 }, then AB ={2, 4, 5}

Union of Sets: The union of A and B is the set containing all elements that are members of A or B or both. This is denoted 'AB.

Example: Given that A = { 2, 3, 4, 5, 6 } and B = { 2, 4,5.,7, 8 }, then AB ={ 2, 3, 4, 5, 6, 7, 8 }

Disjoint Sets: The sets A and B are said to be disjoint if they have no elements in common. Thus,{A} ∩ {B} = ∅.

Complement of a Set: Given a set A, we define its complement, denoted A’, is the set containing all the elements in the universal set S, that are not members of set A. The set A’ is often called “not A”

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  • {A} ∪ { B} ={ 2, 3, 4, 5, 6, 7, 8 }.
  • { 2, 3, 4, 5, 6 } ∪ { 2, 4,5.,7, 8} ={ 2, 3, 4, 5, 6, 7, 8 }.