Which probability is affected by dependent events- P(A) or P(A|C)?Intuitive Understanding of Dependent Events in ProbabilityWhat is the meaning of “independent events ” and how can we logically conclude independence of two events in probability?Probability of independent & mutually exclusive eventsI can't grasp how these events are independent?Independence intuitionProbability of joint events which are dependentDependent eventsAre all disjoint events dependent?Independent Events Conceptual Meaning - probability theoryDetermining whether two events are independent or dependent.
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Which probability is affected by dependent events- P(A) or P(A|C)?
Intuitive Understanding of Dependent Events in ProbabilityWhat is the meaning of “independent events ” and how can we logically conclude independence of two events in probability?Probability of independent & mutually exclusive eventsI can't grasp how these events are independent?Independence intuitionProbability of joint events which are dependentDependent eventsAre all disjoint events dependent?Independent Events Conceptual Meaning - probability theoryDetermining whether two events are independent or dependent.
$begingroup$
Def1. INDEPENDENT EVENTS: two events A and B are independent if occurrence of B does not affect the probability of occurrence of A (and vice versa). [A-ind-B in short]
Def2. DEPENDENT EVENTS: two events A and C are dependent if occurrence of C affects the probability of occurrence of the A (and vice versa). [A-dep-C in short]
Let us have A-ind-B and A-dep-C.
So we get:
N1. B doesn't affect P(A) by above definition.
N2. C affects P(A) by above definition.
But in reality, even if C occurs, it has nothing to do with probability of occurrence of A, P(A).
In other words even though A-dep-C , C still doesn't affect P(A).
So isn't A and C independent by Def1 (because C doesn't affect P(A) )? But A and C are NOT independent, they are dependent (we have assigned them to be dependent).
Let's change the definition of Dependent events:
Def3. DEPENDENT EVENTS: two events A and C are dependent if occurrence of C affects the probability of 'occurrence of A such that C has occurred' (i.e. P(A|C) ) (and vice versa).
In other words, if A-dep-C , then occurrence of C affects P(A|C) (and not P(A) )
So if we compare Def2 and Def3 , this means "probability of occurrence of A" (as in Def2) actually refers to P(A|C) and NOT to P(A). That is P(A|C) is affected, and not P(A).
1. So which one is affected- P(A) or P(A|C) ?
2. And which one should be definition of dependent events- Def2 or Def3 ?
probability conditional-probability independence
asked Mar 29 at 10:34
Ane SaAne Sa
62
$endgroup$
|
show 1 more comment
$begingroup$
Def1. INDEPENDENT EVENTS: two events A and B are independent if occurrence of B does not affect the probability of occurrence of A (and vice versa). [A-ind-B in short]
Def2. DEPENDENT EVENTS: two events A and C are dependent if occurrence of C affects the probability of occurrence of the A (and vice versa). [A-dep-C in short]
Let us have A-ind-B and A-dep-C.
So we get:
N1. B doesn't affect P(A) by above definition.
N2. C affects P(A) by above definition.
But in reality, even if C occurs, it has nothing to do with probability of occurrence of A, P(A).
In other words even though A-dep-C , C still doesn't affect P(A).
So isn't A and C independent by Def1 (because C doesn't affect P(A) )? But A and C are NOT independent, they are dependent (we have assigned them to be dependent).
Let's change the definition of Dependent events:
Def3. DEPENDENT EVENTS: two events A and C are dependent if occurrence of C affects the probability of 'occurrence of A such that C has occurred' (i.e. P(A|C) ) (and vice versa).
In other words, if A-dep-C , then occurrence of C affects P(A|C) (and not P(A) )
So if we compare Def2 and Def3 , this means "probability of occurrence of A" (as in Def2) actually refers to P(A|C) and NOT to P(A). That is P(A|C) is affected, and not P(A).
1. So which one is affected- P(A) or P(A|C) ?
2. And which one should be definition of dependent events- Def2 or Def3 ?
probability conditional-probability independence
asked Mar 29 at 10:34
Ane SaAne Sa
62
$endgroup$
$begingroup$
This is not clear. It is not generally true that "in reality, even if C occurs, it has nothing to do with probability of occurrence of $A$, $P(A)$. In other words even though A-dep-C , C still doesn't affect P(A)." Say you are tossing a fair coin, $A$ is the event that it comes up $H$ and $C$ is the event that it comes up $T$. Then $P(A)=.5$ but $P(A,|,C)=0$. Informally, knowing that $C$ has occurred excludes the chance that $A$ also occurred.
$endgroup$
– lulu
Mar 29 at 11:01
$begingroup$
The wiki article is, I think, quite clear on the definition.
$endgroup$
– lulu
Mar 29 at 11:04
$begingroup$
Keep in mind: informal definitions are just that, informal. They serve a purpose in that they communicate the intent of the definition in simple terms. Usually they are not good mathematical definitions. They use imprecise, ill-defined terms. In order to be useful mathematically they must be accompanied with a formal definition. The wiki article I linked to does that...it starts with the "ordinary language" definition which, as you remark, is imprecise....
$endgroup$
– lulu
Mar 29 at 11:10
$begingroup$
...But then it supplies the formal definition...$A,B$ are independent if $P(Acap B)=P(A)times P(B)$. They then show that this is equivalent to $P(A)=P(A,|,B)$ which, I'd say, is the best mathematical translation of the informal definition.
$endgroup$
– lulu
Mar 29 at 11:10
$begingroup$
@lulu thanks for your help. The word definition or informal definition really confused me while studying independent events. You explaination really helped me and you made me understand the concept better. Thanks for help. $_/backslash_$
$endgroup$
– Ane Sa
Mar 29 at 21:54
|
show 1 more comment
$begingroup$
Def1. INDEPENDENT EVENTS: two events A and B are independent if occurrence of B does not affect the probability of occurrence of A (and vice versa). [A-ind-B in short]
Def2. DEPENDENT EVENTS: two events A and C are dependent if occurrence of C affects the probability of occurrence of the A (and vice versa). [A-dep-C in short]
Let us have A-ind-B and A-dep-C.
So we get:
N1. B doesn't affect P(A) by above definition.
N2. C affects P(A) by above definition.
But in reality, even if C occurs, it has nothing to do with probability of occurrence of A, P(A).
In other words even though A-dep-C , C still doesn't affect P(A).
So isn't A and C independent by Def1 (because C doesn't affect P(A) )? But A and C are NOT independent, they are dependent (we have assigned them to be dependent).
Let's change the definition of Dependent events:
Def3. DEPENDENT EVENTS: two events A and C are dependent if occurrence of C affects the probability of 'occurrence of A such that C has occurred' (i.e. P(A|C) ) (and vice versa).
In other words, if A-dep-C , then occurrence of C affects P(A|C) (and not P(A) )
So if we compare Def2 and Def3 , this means "probability of occurrence of A" (as in Def2) actually refers to P(A|C) and NOT to P(A). That is P(A|C) is affected, and not P(A).
1. So which one is affected- P(A) or P(A|C) ?
2. And which one should be definition of dependent events- Def2 or Def3 ?
probability conditional-probability independence
asked Mar 29 at 10:34
Ane SaAne Sa
62
$endgroup$
Def1. INDEPENDENT EVENTS: two events A and B are independent if occurrence of B does not affect the probability of occurrence of A (and vice versa). [A-ind-B in short]
Def2. DEPENDENT EVENTS: two events A and C are dependent if occurrence of C affects the probability of occurrence of the A (and vice versa). [A-dep-C in short]
Let us have A-ind-B and A-dep-C.
So we get:
N1. B doesn't affect P(A) by above definition.
N2. C affects P(A) by above definition.
But in reality, even if C occurs, it has nothing to do with probability of occurrence of A, P(A).
In other words even though A-dep-C , C still doesn't affect P(A).
So isn't A and C independent by Def1 (because C doesn't affect P(A) )? But A and C are NOT independent, they are dependent (we have assigned them to be dependent).
Let's change the definition of Dependent events:
Def3. DEPENDENT EVENTS: two events A and C are dependent if occurrence of C affects the probability of 'occurrence of A such that C has occurred' (i.e. P(A|C) ) (and vice versa).
In other words, if A-dep-C , then occurrence of C affects P(A|C) (and not P(A) )
So if we compare Def2 and Def3 , this means "probability of occurrence of A" (as in Def2) actually refers to P(A|C) and NOT to P(A). That is P(A|C) is affected, and not P(A).
1. So which one is affected- P(A) or P(A|C) ?
2. And which one should be definition of dependent events- Def2 or Def3 ?
probability conditional-probability independence
probability conditional-probability independence
asked Mar 29 at 10:34
Ane SaAne Sa
62
asked Mar 29 at 10:34
Ane SaAne Sa
62
edited Mar 29 at 21:41
Ane Sa
asked Mar 29 at 10:34
Ane SaAne Sa
62
asked Mar 29 at 10:34
Ane SaAne Sa
62
asked Mar 29 at 10:34
Ane SaAne Sa
62
62
$begingroup$
This is not clear. It is not generally true that "in reality, even if C occurs, it has nothing to do with probability of occurrence of $A$, $P(A)$. In other words even though A-dep-C , C still doesn't affect P(A)." Say you are tossing a fair coin, $A$ is the event that it comes up $H$ and $C$ is the event that it comes up $T$. Then $P(A)=.5$ but $P(A,|,C)=0$. Informally, knowing that $C$ has occurred excludes the chance that $A$ also occurred.
$endgroup$
– lulu
Mar 29 at 11:01
$begingroup$
The wiki article is, I think, quite clear on the definition.
$endgroup$
– lulu
Mar 29 at 11:04
$begingroup$
Keep in mind: informal definitions are just that, informal. They serve a purpose in that they communicate the intent of the definition in simple terms. Usually they are not good mathematical definitions. They use imprecise, ill-defined terms. In order to be useful mathematically they must be accompanied with a formal definition. The wiki article I linked to does that...it starts with the "ordinary language" definition which, as you remark, is imprecise....
$endgroup$
– lulu
Mar 29 at 11:10
$begingroup$
...But then it supplies the formal definition...$A,B$ are independent if $P(Acap B)=P(A)times P(B)$. They then show that this is equivalent to $P(A)=P(A,|,B)$ which, I'd say, is the best mathematical translation of the informal definition.
$endgroup$
– lulu
Mar 29 at 11:10
$begingroup$
@lulu thanks for your help. The word definition or informal definition really confused me while studying independent events. You explaination really helped me and you made me understand the concept better. Thanks for help. $_/backslash_$
$endgroup$
– Ane Sa
Mar 29 at 21:54
|
show 1 more comment
$begingroup$
This is not clear. It is not generally true that "in reality, even if C occurs, it has nothing to do with probability of occurrence of $A$, $P(A)$. In other words even though A-dep-C , C still doesn't affect P(A)." Say you are tossing a fair coin, $A$ is the event that it comes up $H$ and $C$ is the event that it comes up $T$. Then $P(A)=.5$ but $P(A,|,C)=0$. Informally, knowing that $C$ has occurred excludes the chance that $A$ also occurred.
$endgroup$
– lulu
Mar 29 at 11:01
$begingroup$
The wiki article is, I think, quite clear on the definition.
$endgroup$
– lulu
Mar 29 at 11:04
$begingroup$
Keep in mind: informal definitions are just that, informal. They serve a purpose in that they communicate the intent of the definition in simple terms. Usually they are not good mathematical definitions. They use imprecise, ill-defined terms. In order to be useful mathematically they must be accompanied with a formal definition. The wiki article I linked to does that...it starts with the "ordinary language" definition which, as you remark, is imprecise....
$endgroup$
– lulu
Mar 29 at 11:10
$begingroup$
...But then it supplies the formal definition...$A,B$ are independent if $P(Acap B)=P(A)times P(B)$. They then show that this is equivalent to $P(A)=P(A,|,B)$ which, I'd say, is the best mathematical translation of the informal definition.
$endgroup$
– lulu
Mar 29 at 11:10
$begingroup$
@lulu thanks for your help. The word definition or informal definition really confused me while studying independent events. You explaination really helped me and you made me understand the concept better. Thanks for help. $_/backslash_$
$endgroup$
– Ane Sa
Mar 29 at 21:54
$begingroup$
This is not clear. It is not generally true that "in reality, even if C occurs, it has nothing to do with probability of occurrence of $A$, $P(A)$. In other words even though A-dep-C , C still doesn't affect P(A)." Say you are tossing a fair coin, $A$ is the event that it comes up $H$ and $C$ is the event that it comes up $T$. Then $P(A)=.5$ but $P(A,|,C)=0$. Informally, knowing that $C$ has occurred excludes the chance that $A$ also occurred.
$endgroup$
– lulu
Mar 29 at 11:01
$begingroup$
This is not clear. It is not generally true that "in reality, even if C occurs, it has nothing to do with probability of occurrence of $A$, $P(A)$. In other words even though A-dep-C , C still doesn't affect P(A)." Say you are tossing a fair coin, $A$ is the event that it comes up $H$ and $C$ is the event that it comes up $T$. Then $P(A)=.5$ but $P(A,|,C)=0$. Informally, knowing that $C$ has occurred excludes the chance that $A$ also occurred.
$endgroup$
– lulu
Mar 29 at 11:01
$begingroup$
The wiki article is, I think, quite clear on the definition.
$endgroup$
– lulu
Mar 29 at 11:04
$begingroup$
The wiki article is, I think, quite clear on the definition.
$endgroup$
– lulu
Mar 29 at 11:04
$begingroup$
Keep in mind: informal definitions are just that, informal. They serve a purpose in that they communicate the intent of the definition in simple terms. Usually they are not good mathematical definitions. They use imprecise, ill-defined terms. In order to be useful mathematically they must be accompanied with a formal definition. The wiki article I linked to does that...it starts with the "ordinary language" definition which, as you remark, is imprecise....
$endgroup$
– lulu
Mar 29 at 11:10
$begingroup$
Keep in mind: informal definitions are just that, informal. They serve a purpose in that they communicate the intent of the definition in simple terms. Usually they are not good mathematical definitions. They use imprecise, ill-defined terms. In order to be useful mathematically they must be accompanied with a formal definition. The wiki article I linked to does that...it starts with the "ordinary language" definition which, as you remark, is imprecise....
$endgroup$
– lulu
Mar 29 at 11:10
$begingroup$
...But then it supplies the formal definition...$A,B$ are independent if $P(Acap B)=P(A)times P(B)$. They then show that this is equivalent to $P(A)=P(A,|,B)$ which, I'd say, is the best mathematical translation of the informal definition.
$endgroup$
– lulu
Mar 29 at 11:10
$begingroup$
...But then it supplies the formal definition...$A,B$ are independent if $P(Acap B)=P(A)times P(B)$. They then show that this is equivalent to $P(A)=P(A,|,B)$ which, I'd say, is the best mathematical translation of the informal definition.
$endgroup$
– lulu
Mar 29 at 11:10
$begingroup$
@lulu thanks for your help. The word definition or informal definition really confused me while studying independent events. You explaination really helped me and you made me understand the concept better. Thanks for help. $_/backslash_$
$endgroup$
– Ane Sa
Mar 29 at 21:54
$begingroup$
@lulu thanks for your help. The word definition or informal definition really confused me while studying independent events. You explaination really helped me and you made me understand the concept better. Thanks for help. $_/backslash_$
$endgroup$
– Ane Sa
Mar 29 at 21:54
|
show 1 more comment
0
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$begingroup$
This is not clear. It is not generally true that "in reality, even if C occurs, it has nothing to do with probability of occurrence of $A$, $P(A)$. In other words even though A-dep-C , C still doesn't affect P(A)." Say you are tossing a fair coin, $A$ is the event that it comes up $H$ and $C$ is the event that it comes up $T$. Then $P(A)=.5$ but $P(A,|,C)=0$. Informally, knowing that $C$ has occurred excludes the chance that $A$ also occurred.
$endgroup$
– lulu
Mar 29 at 11:01
$begingroup$
The wiki article is, I think, quite clear on the definition.
$endgroup$
– lulu
Mar 29 at 11:04
$begingroup$
Keep in mind: informal definitions are just that, informal. They serve a purpose in that they communicate the intent of the definition in simple terms. Usually they are not good mathematical definitions. They use imprecise, ill-defined terms. In order to be useful mathematically they must be accompanied with a formal definition. The wiki article I linked to does that...it starts with the "ordinary language" definition which, as you remark, is imprecise....
$endgroup$
– lulu
Mar 29 at 11:10
$begingroup$
...But then it supplies the formal definition...$A,B$ are independent if $P(Acap B)=P(A)times P(B)$. They then show that this is equivalent to $P(A)=P(A,|,B)$ which, I'd say, is the best mathematical translation of the informal definition.
$endgroup$
– lulu
Mar 29 at 11:10
$begingroup$
@lulu thanks for your help. The word definition or informal definition really confused me while studying independent events. You explaination really helped me and you made me understand the concept better. Thanks for help. $_/backslash_$
$endgroup$
– Ane Sa
Mar 29 at 21:54