Question

if \(A \cap B\), \(A' \cap B\), \(B \cap A'\) are independent events, how to prove that \(A' \cap B'\) are also independent?

I know that they are independent but I don't know how to prove.

Answer 1

I think you need to clarify the problem. it doesn't look right

Answer 2

one set is not "independent"

Answer 3

What do we mean by independent here?

Answer 4

you need two sets. A and B are independent means
\[P(A|B)=P(A)\]

Answer 5

or if you prefer
\[\frac{P(A\cap B)}{P(B)}=P(A)\]

Answer 6

it is probably best to write

\[P(A\cap B)=P(A)P(B)\]

just in case \(P(B)=0\)

Answer 7

we do have this...if A and B are independent then

A and B' are independent
A' and B are independent
A' and B' are independent


here the 'and' is not the intersection

Answer 8

yes i suppose. then again if P(B) = 0 then A and B are independent for all A

Answer 9

i am going to bet that zarkon stated the question correctly.

Answer 10

\[P(A'\cap B')\]
\[=1-P(A\cup B)\]
\[=1-(P(A)+P(B)-P(A\cap B))\]
\[=1-P(A)-P(B)+P(A\cap B)\]
\[=1-P(A)-P(B)+P(A)P(B)\]
\[=1-P(A)-P(B)(1-P(A))\]
\[=(1-P(A))(1-P(B))\]
\[=P(A')P(B')\]

Answer 11

yeah zarkon stated it correctly lol

Answer 12

sorry for that