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

Independency of Events

Three events  \text { A,B } and  \text{ C } are independent if and only if all the following conditions hold:

(i)  P(A\cap B)=P(A)\cdot P(B)

(ii)  P(B\cap C)=P(B)\cdot P(C)

(iii)  P(C\cap A)=P(C)\cdot P(A)

(iv)  P(A\cap B\cap C)=P(A)\cdot P(B)\cdot P(C)

i.e., they must be independent in pairs as well as mutually independent.

→For  n events  \operatorname {A_1,A_2,...,A_n} to be independent, the number of these conditions is equal to  ^nC_2+^nC_3+...+^nC_n=2^n-n-1

Laws of Total Probability

If an event  \text {A} can occur with one of the  n mutually exclusive and exhaustive events  \text{B_1,B_2,...,B_n } and the probabilities  P\left(\dfrac{A}{B}\right),P\left(\dfrac{A}{B_2}\right),...,P\left(\dfrac{A}{B_n}\right) are known, then  P(A)=\sum_{i=1}^{n}P(B_i)\cdot P\left(\dfrac{A}{B_i}\right)

Note:

For any mutually exclusive distinct  \text{A_1,A_2,...,A_n,...} then

 P\left( \bigcup_{i=1}^{\infty }A_i \right)=\sum_{i=1}^{\infty }P(A_i)

 \implies P(A_1\cup A_2\Cup...A_n\cup...)=P(A_i)+P(A_2)+...+P(A_n)+...

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