Mutual independence of three events The Next CEO of Stack OverflowUnderstanding dependency graph for a set of eventsWhat is the meaning of “independent events ” and how can we logically conclude independence of two events in probability?Prove not mutually independentWhy are the probability of rolling the same number twice and the probability of rolling pairs different?Proving pairwise independence and mutually independentWhy are events $A$ and $B$ independent whilst events $A$ and $C$ are not?Independence and Mutual Exclusion of a Continuous Random VariableVariance of mutually exclusive eventsIndependence of random variables and eventsIntuition of mutually independent events
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Mutual independence of three events
The Next CEO of Stack OverflowUnderstanding dependency graph for a set of eventsWhat is the meaning of “independent events ” and how can we logically conclude independence of two events in probability?Prove not mutually independentWhy are the probability of rolling the same number twice and the probability of rolling pairs different?Proving pairwise independence and mutually independentWhy are events $A$ and $B$ independent whilst events $A$ and $C$ are not?Independence and Mutual Exclusion of a Continuous Random VariableVariance of mutually exclusive eventsIndependence of random variables and eventsIntuition of mutually independent events
$begingroup$
Is it possible to have three events $A,B,C$ such that $A$ is mutually independent to $B,C$ and $B$ is NOT mutually independent to $A,C$.
By mutual independence I mean, $A$ is mutually independent to a set $S$ if $$Pleft[Aleft|bigcap_Xin S Xright.right] = P[A]$$ for all subsets $X$ of $S$.
I tried thinking of it in terms of rolling two dice. Having $A,B$ be the events such that the first die results in something, and the second die results in something else. Also having $C$ be the sum of the two dice resulting in some value. But no matter what events I choose for $A,B,C$ I can't come up with ones that fit my constraints. Any help would be appreciated.
probability independence
asked Mar 18 at 19:50
JohnnyJohnny
63
$endgroup$
add a comment |
$begingroup$
Is it possible to have three events $A,B,C$ such that $A$ is mutually independent to $B,C$ and $B$ is NOT mutually independent to $A,C$.
By mutual independence I mean, $A$ is mutually independent to a set $S$ if $$Pleft[Aleft|bigcap_Xin S Xright.right] = P[A]$$ for all subsets $X$ of $S$.
I tried thinking of it in terms of rolling two dice. Having $A,B$ be the events such that the first die results in something, and the second die results in something else. Also having $C$ be the sum of the two dice resulting in some value. But no matter what events I choose for $A,B,C$ I can't come up with ones that fit my constraints. Any help would be appreciated.
probability independence
asked Mar 18 at 19:50
JohnnyJohnny
63
$endgroup$
add a comment |
$begingroup$
Is it possible to have three events $A,B,C$ such that $A$ is mutually independent to $B,C$ and $B$ is NOT mutually independent to $A,C$.
By mutual independence I mean, $A$ is mutually independent to a set $S$ if $$Pleft[Aleft|bigcap_Xin S Xright.right] = P[A]$$ for all subsets $X$ of $S$.
I tried thinking of it in terms of rolling two dice. Having $A,B$ be the events such that the first die results in something, and the second die results in something else. Also having $C$ be the sum of the two dice resulting in some value. But no matter what events I choose for $A,B,C$ I can't come up with ones that fit my constraints. Any help would be appreciated.
probability independence
asked Mar 18 at 19:50
JohnnyJohnny
63
$endgroup$
Is it possible to have three events $A,B,C$ such that $A$ is mutually independent to $B,C$ and $B$ is NOT mutually independent to $A,C$.
By mutual independence I mean, $A$ is mutually independent to a set $S$ if $$Pleft[Aleft|bigcap_Xin S Xright.right] = P[A]$$ for all subsets $X$ of $S$.
I tried thinking of it in terms of rolling two dice. Having $A,B$ be the events such that the first die results in something, and the second die results in something else. Also having $C$ be the sum of the two dice resulting in some value. But no matter what events I choose for $A,B,C$ I can't come up with ones that fit my constraints. Any help would be appreciated.
probability independence
probability independence
asked Mar 18 at 19:50
JohnnyJohnny
63
asked Mar 18 at 19:50
JohnnyJohnny
63
edited Mar 18 at 19:59
Johnny
asked Mar 18 at 19:50
JohnnyJohnny
63
asked Mar 18 at 19:50
JohnnyJohnny
63
asked Mar 18 at 19:50
JohnnyJohnny
63
63
add a comment |
add a comment |
2 Answers
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$begingroup$
$A$ being mutually independent of $B$ means $P(Acap B) / P(B) = P(A)$.
That means $P(Acap B) = P(B) P(A)$, so if $A$ is mutually independent of $B$, then $B$ is mutually independent of $A$. You can make $A$, $B$ two rolls of two distinct dice, and $C$ to be the exact same event as $B$. Then $A$ is independent to $B,C$, and $B$ is independent to $A$, but $B$ is not independent to $C$, because $B$ is $C$.
answered Mar 18 at 20:16
user3257842user3257842
1569
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add a comment |
$begingroup$
Yes, this is possible. One example would be the case that $B,C$ refer to the same event, and $A$ is independent of $B$. Then $A$ is mutually independent to $B,C$ but $B$ is not mutually independent to $A,C$.
You can easily construct cases in which $B,C$ are not identical events. E.g., $A$ is the event that die 1 comes up even, $B$ is the event that die 2 comes up odd, $C$ is the event that die 2 comes up prime.
answered Mar 18 at 20:21
CephCeph
874415
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add a comment |
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2 Answers
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2 Answers
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$begingroup$
$A$ being mutually independent of $B$ means $P(Acap B) / P(B) = P(A)$.
That means $P(Acap B) = P(B) P(A)$, so if $A$ is mutually independent of $B$, then $B$ is mutually independent of $A$. You can make $A$, $B$ two rolls of two distinct dice, and $C$ to be the exact same event as $B$. Then $A$ is independent to $B,C$, and $B$ is independent to $A$, but $B$ is not independent to $C$, because $B$ is $C$.
answered Mar 18 at 20:16
user3257842user3257842
1569
$endgroup$
add a comment |
$begingroup$
$A$ being mutually independent of $B$ means $P(Acap B) / P(B) = P(A)$.
That means $P(Acap B) = P(B) P(A)$, so if $A$ is mutually independent of $B$, then $B$ is mutually independent of $A$. You can make $A$, $B$ two rolls of two distinct dice, and $C$ to be the exact same event as $B$. Then $A$ is independent to $B,C$, and $B$ is independent to $A$, but $B$ is not independent to $C$, because $B$ is $C$.
answered Mar 18 at 20:16
user3257842user3257842
1569
$endgroup$
add a comment |
$begingroup$
$A$ being mutually independent of $B$ means $P(Acap B) / P(B) = P(A)$.
That means $P(Acap B) = P(B) P(A)$, so if $A$ is mutually independent of $B$, then $B$ is mutually independent of $A$. You can make $A$, $B$ two rolls of two distinct dice, and $C$ to be the exact same event as $B$. Then $A$ is independent to $B,C$, and $B$ is independent to $A$, but $B$ is not independent to $C$, because $B$ is $C$.
answered Mar 18 at 20:16
user3257842user3257842
1569
$endgroup$
$A$ being mutually independent of $B$ means $P(Acap B) / P(B) = P(A)$.
That means $P(Acap B) = P(B) P(A)$, so if $A$ is mutually independent of $B$, then $B$ is mutually independent of $A$. You can make $A$, $B$ two rolls of two distinct dice, and $C$ to be the exact same event as $B$. Then $A$ is independent to $B,C$, and $B$ is independent to $A$, but $B$ is not independent to $C$, because $B$ is $C$.
answered Mar 18 at 20:16
user3257842user3257842
1569
answered Mar 18 at 20:16
user3257842user3257842
1569
answered Mar 18 at 20:16
user3257842user3257842
1569
answered Mar 18 at 20:16
user3257842user3257842
1569
1569
add a comment |
add a comment |
$begingroup$
Yes, this is possible. One example would be the case that $B,C$ refer to the same event, and $A$ is independent of $B$. Then $A$ is mutually independent to $B,C$ but $B$ is not mutually independent to $A,C$.
You can easily construct cases in which $B,C$ are not identical events. E.g., $A$ is the event that die 1 comes up even, $B$ is the event that die 2 comes up odd, $C$ is the event that die 2 comes up prime.
answered Mar 18 at 20:21
CephCeph
874415
$endgroup$
add a comment |
$begingroup$
Yes, this is possible. One example would be the case that $B,C$ refer to the same event, and $A$ is independent of $B$. Then $A$ is mutually independent to $B,C$ but $B$ is not mutually independent to $A,C$.
You can easily construct cases in which $B,C$ are not identical events. E.g., $A$ is the event that die 1 comes up even, $B$ is the event that die 2 comes up odd, $C$ is the event that die 2 comes up prime.
answered Mar 18 at 20:21
CephCeph
874415
$endgroup$
add a comment |
$begingroup$
Yes, this is possible. One example would be the case that $B,C$ refer to the same event, and $A$ is independent of $B$. Then $A$ is mutually independent to $B,C$ but $B$ is not mutually independent to $A,C$.
You can easily construct cases in which $B,C$ are not identical events. E.g., $A$ is the event that die 1 comes up even, $B$ is the event that die 2 comes up odd, $C$ is the event that die 2 comes up prime.
answered Mar 18 at 20:21
CephCeph
874415
$endgroup$
Yes, this is possible. One example would be the case that $B,C$ refer to the same event, and $A$ is independent of $B$. Then $A$ is mutually independent to $B,C$ but $B$ is not mutually independent to $A,C$.
You can easily construct cases in which $B,C$ are not identical events. E.g., $A$ is the event that die 1 comes up even, $B$ is the event that die 2 comes up odd, $C$ is the event that die 2 comes up prime.
answered Mar 18 at 20:21
CephCeph
874415
answered Mar 18 at 20:21
CephCeph
874415
answered Mar 18 at 20:21
CephCeph
874415
answered Mar 18 at 20:21
CephCeph
874415
874415
add a comment |
add a comment |
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