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Let A, B and C be three events, which are pair-wise independent and E denotes the
complement of an event E. If P(A\cap B\cap C)=0  andP(C)> 0 then 

P[(\bar{A}\cap \bar{B}|C)] is equal to

  • Option 1)

    P(\bar{A)}-P(B)

  • Option 2)

    P(A)+P(\bar{B)}

  • Option 3)

    P(\bar{A)}-P(\bar B)

     

  • Option 4)

    P(\bar{A)}+P(\bar B)

 

Answers (1)

best_answer

As we have learned

Independent events -

If A and B are independent events then A and \overline{B} as well as \overline{A} and B are independent events. 

- wherein

 

 

Addition Theorem of Probability -

P\left ( A\cup B \right )= P\left ( A \right )+P\left ( B \right )-P\left ( A\cap B \right )

in general:

P\left ( A_{1}\cup A_{2}\cup A_{3}\cdots A_{n} \right )=\sum_{i=1 }^{n}P\left ( A_{i} \right )-\sum_{i< j}^{n}P\left ( A_{i}\cap A_{j} \right )+\sum_{i< j< k}^{n} P\left ( A_{i}\cap A_{j}\cap A_{k} \right )-\cdots -\left ( -1 \right )^{n-1}P\left ( A_{1}\cap A_{2}\cap A_{3}\cdots \cap A_{n} \right )

-

 

 p(A\cap B)= p(A)\cdot p(B)

also  \frac{p(A\cap B\cap C)}{p(C)}

=p(C) -\frac{p(C\cap A)}{p(C)}-p(C\cap B) + p(C\cap A\cap B)

= 1-p(A)-p(B)

 

=p(\bar{A})-p(B)

 

 

 

 

 

 

 

 


Option 1)

P(\bar{A)}-P(B)

This is correct

Option 2)

P(A)+P(\bar{B)}

This is incorrect

Option 3)

P(\bar{A)}-P(\bar B)

 

This is incorrect

Option 4)

P(\bar{A)}+P(\bar B)

This is incorrect

Posted by

Himanshu

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