Complement Proof of independent events

**dwsmith** · February 3rd 2011, 04:51 PM

If A and B are independent events, show that $\bar{A} \ \ \text{and} \ \ \bar{B}$ are independent.

I haven't a clue what to do for this one.

**harish21** · February 3rd 2011, 05:10 PM

Originally Posted by dwsmith

If A and B are independent events, show that $\bar{A} \ \ \text{and} \ \ \bar{B}$ are independent.

I haven't a clue what to do for this one.

I usually write $A^c$ for complements. You can show $P(A^c \cap B^c) = P(A^c) P(B^c)$ .

$P(A^c \cap B^c)$

$= P[(A \cup B)^c]$

$= 1-P(A \cup B)$

$= 1-P(A)-P(B)+P(A\cap B)$

now use $P(A\cap B) = P(A) P(B)\; and\; P(A^c)=1-P(A)\;and\;P(B^c)=1-P(B)$ and simplify

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