Thread: Show event is completly independent of another event

1. Show event is completly independent of another event

Question :
Show that if the event A is completely independent of events B and C, then A is independent of logical sum $B \cup C$

2. assume that

$P(AB)=P(A)P(B)$ and $P(AC)=P(A)P(C)$

the question is, does

$P(A(B\cup C))=P(A)P(B\cup C)$????????

I don't think so.
Try tossing a fair die.
Let $A=2,4,6$
$B=2,3$
and
$C=2,5$

I get 1/6 =1/6 for the first two
but then I get 1/6 and 1/4 on the last one.

3. Could u please explain me

Originally Posted by matheagle
assume that

$P(AB)=P(A)P(B)$ and $P(AC)=P(A)P(C)$

the question is, does

$P(A(B\cup C))=P(A)P(B\cup C)$????????

I don't think so.
Try tossing a fair die.
Let $A=2,4,6$
$B=2,3$
and
$C=2,5$

I get 1/6 =1/6 for the first two
but then I get 1/6 and 1/4 on the last one.

could u please explain a little bit properly?/

4. Originally Posted by zorro
Question :
Show that if the event A is completely independent of events B and C, then A is independent of logical sum $B \cup C$
I guess "completely independent" is what is also called "mutually independent" (by opposition to "pairwise independent"), i.e. it means that $P(A\cap B)=P(A)P(B)$ (same with A,C and B,C), and $P(A\cap B\cap C)=P(A)P(B)P(C)$.

Then the property is proved as follows (for instance):
$P(A\cap(B\cup C))=P((A\cap B\cap C)\cup(A\cap B^c\cap C)\cup (A\cap B\cap C^c))$, and the three (main) events on the right-hand side are disjoints, hence
$P(A\cap(B\cup C))=P(A\cap B\cap C)+P(A\cap B^c\cap C)+P(A\cap B\cap C^c)$,
and you can compute the latter probabilities using $A\cap B^c\cap C=(A\cap C)\setminus (A\cap B\cap C)$ (and similarly for the last term) and the assumptions (recalled above).

You should get $\cdots=P(A)P(B\cup C)$, or in fact $P(A)(P(B)+P(C)-P(B)P(C))$, which is the same...

5. I am again stuck

Originally Posted by Laurent
I guess "completely independent" is what is also called "mutually independent" (by opposition to "pairwise independent"), i.e. it means that $P(A\cap B)=P(A)P(B)$ (same with A,C and B,C), and $P(A\cap B\cap C)=P(A)P(B)P(C)$.

Then the property is proved as follows (for instance):
$P(A\cap(B\cup C))=P((A\cap B\cap C)\cup(A\cap B^c\cap C)\cup (A\cap B\cap C^c))$, and the three (main) events on the right-hand side are disjoints, hence
$P(A\cap(B\cup C))=P(A\cap B\cap C)+P(A\cap B^c\cap C)+P(A\cap B\cap C^c)$,
and you can compute the latter probabilities using $A\cap B^c\cap C=(A\cap C)\setminus (A\cap B\cap C)$ (and similarly for the last term) and the assumptions (recalled above).

You should get $\cdots=P(A)P(B\cup C)$, or in fact $P(A)(P(B)+P(C)-P(B)P(C))$, which is the same...

,

$
Pr(A \cap(B \cup C)) = Pr(A \cap B \cap C) + \frac{Pr(A \cap C)}{Pr(A \cap B \cap C)} + \frac{Pr(A \cap B)}{Pr(A \cap B \cap C)}$

I am stuck here ????

6. Originally Posted by zorro
,

$
Pr(A \cap(B \cup C)) = Pr(A \cap B \cap C) + \frac{Pr(A \cap C)}{Pr(A \cap B \cap C)} + \frac{Pr(A \cap B)}{Pr(A \cap B \cap C)}$

I am stuck here ????

If $B\subset A$, $P(A\setminus B)=P(A)-P(B)$, of course... not $\frac{P(A)}{P(B)}$...

And then just use the assumption of independence.

7. I did get u

Originally Posted by Laurent

If $B\subset A$, $P(A\setminus B)=P(A)-P(B)$, of course... not $\frac{P(A)}{P(B)}$...

And then just use the assumption of independence.