Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Equiprobable Space or Uniform Space

A sample space  \text{ S } is called an Equiprobable space iff all the simple events are equally likely to occur:

→Probability of occurrence of the event  \text{ A } , donated by  \text{P(A)} , is defined by

 \begin{aligned} P(A)&=\dfrac{\text{Size of A}}{\text{Size of S}}\\ &=\dfrac{\text{n(A)}}{\text{n(S)}} \end{aligned}

where  \text{'S'} is the Sample Space.

→Probability of non-occurrence of  \text{A=1-P(A)}

Algebra of Events:

(i) For any event  A,0  P(A)  1 .

(ii)  \text{P(S)}=1

(iii)  \text{P}(\phi)=0 , where  \phi is empty set.

(iv)  \text{P(A')=1-P(A)}

(v) If  \text{A,B} are two events and  A\underline{C}B , then \text{ P(A)}  \text{P(B)}

(vi) If  \text{A,B} are any two events, then  P(A\cup B) =P(A)+P(B)-P(A\cap B)

(vii) If  \text{A,B} are mutually exclusive events, that is  A\cap B=\phi , then  P(A\cup B)=P(A)+P(B)

(viii)  P(A-B)=P(A)-P(A\cup B) or  P(A-B)=P(A\cup)-P(B)

(ix)  P(B-A)=P(B)-P(A\cup B) or  P(B-A)=P(A\cup B)-P(A)

(x) If  \text{A,B,C} are events, then:

 P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(B\cap C)-P(C\cap A)+P(A\cap B\cap C)

(xi) For  \text{A,B,C} are mutually exclusive events  P(A\cup B\cup C)=P(A)+P(B)+P(C)

(xii)  \text{ P} ( at least two of  \text{ A,B,C} occur)  =P(B\cap C)+P(C\cap A)+P(A\cap B)-2P(A\cap B\cap C)

(xiii)  \text{ P} ( Exactly two of  \text{ A,B,C } occur)  =P(B\cap C)+P(C\cap A)+P(A\cap B)-3P(A\cap B\cap C)

(xiv)  \text{ P} ( Exactly one of  \text{ A,B,C } occur )

 =P(A)+P(B)+P(C)-2P(B\cap C)-2P(C\cap A)-2P(A\cap B)+3P(A\cap B\cap C)

Note:

If three events  \text{ A,B,C} are pairwise mutually exclusive, then they must be mutually exclusive, that is  P(A\cap B)=P(B\cap C)=P(C\cap A)=0\implies P(A\cap B\cap C)=0

Conditional Probability:

If  \text{A} and  \text{B} are two events, then:

(i)  P\left(\dfrac{A}{B}\right)=\dfrac{P(A\cap B)}{P(B)}

(ii)  P\left(\dfrac{B}{A}\right)=\dfrac{P(A\cap B)}{P(A)}

Note:

For mutually exclusive events that is  A\cap B=\phi , then  P\left(\dfrac{A}{B}\right)=0 and  P\left(\dfrac{B}{A}\right)=0

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