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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 $P(A\cap B\cap C)=0$ and $P(C)> 0$ 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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