# Thread: Complement Proof of independent events

1. ## Complement Proof of independent events

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

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

2. Originally Posted by dwsmith
If A and B are independent events, show that $\displaystyle \bar{A} \ \ \text{and} \ \ \bar{B}$ are independent.

I haven't a clue what to do for this one.
I usually write $\displaystyle A^c$ for complements. You can show $\displaystyle P(A^c \cap B^c) = P(A^c) P(B^c)$.

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

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

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

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

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

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# if a and b are independent events show that a and b complement are also independent

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