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Question:
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Let S be the sample space of a random
experiment and P be a probability function
defined on the power set of S. Two events A
and B of the random experiment are called
independent if
Options:
experiment and P be a probability function
defined on the power set of S. Two events A
and B of the random experiment are called
independent if
Solution:
2449 Upvotes
Verified Answer
The correct answer is:
$P\left(A^C \cap B^C\right)=(1-P(A))(1-P(B))$
It is obvious
$\because$ If $A$ and $B$ are independent, then
$P(A \cap B)=P(A) \cdot P(B)$
By given options, option (a), (b) and (d) are rejected.
Method (2)
$\begin{aligned}
& {[1-P(A)][1-P(B)]=1-P(A)-P(B)+P(A) \cdot P(B)} \\
& =1-[P(A)+P(B)-P(A) \cdot P(B)] \\
& =1-[P(A \cup B)] \quad\{\because P(A \cap B)=P(A) \cdot P(B)\} \\
& =P(\overline{A \cup B}) \\
& =P(\bar{A} \cap \bar{B}) \\
& =P\left(A^C \cap B^C\right)
\end{aligned}$
$\because$ If $A$ and $B$ are independent, then
$P(A \cap B)=P(A) \cdot P(B)$
By given options, option (a), (b) and (d) are rejected.
Method (2)
$\begin{aligned}
& {[1-P(A)][1-P(B)]=1-P(A)-P(B)+P(A) \cdot P(B)} \\
& =1-[P(A)+P(B)-P(A) \cdot P(B)] \\
& =1-[P(A \cup B)] \quad\{\because P(A \cap B)=P(A) \cdot P(B)\} \\
& =P(\overline{A \cup B}) \\
& =P(\bar{A} \cap \bar{B}) \\
& =P\left(A^C \cap B^C\right)
\end{aligned}$
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