If A is dependent upon B, is B necessarily dependent upon A?











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I read that independence of two events has a symmetric relationship. Does this relationship hold for dependent events as well?



If so, what would be the intuition?



Thanks










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    up vote
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    down vote

    favorite












    I read that independence of two events has a symmetric relationship. Does this relationship hold for dependent events as well?



    If so, what would be the intuition?



    Thanks










    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I read that independence of two events has a symmetric relationship. Does this relationship hold for dependent events as well?



      If so, what would be the intuition?



      Thanks










      share|cite|improve this question













      I read that independence of two events has a symmetric relationship. Does this relationship hold for dependent events as well?



      If so, what would be the intuition?



      Thanks







      probability probability-theory






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      asked Nov 22 at 13:52









      Kohler Fryer

      1031




      1031






















          2 Answers
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          accepted










          Yes, two events are dependent if and only if they are not independent.
          Since, if $mu$ is a probability measure, then events A and B are independent if and only if
          begin{equation}
          mu(Acap B)=mu(A)cdotmu(B)
          end{equation}

          and dependent otherwise. Since multiplication is commutative you easily get exactly what you're looking for.



          An easy example is a double coin toss: intuitively the result of tossing the first coin should have no effect on the second and vice-versa.






          share|cite|improve this answer





















          • Thanks, any recommendation on becoming more familiar with this stuff? I'm currently taking a class at a University that is covering this content a bit too fast for my liking.
            – Kohler Fryer
            Nov 22 at 14:46










          • I'm a big fan of the MIT open courseware and would thus recommend taking a peek at that and the books they used. This also depends on the flavour of probility you are doing. For a more statistical/applied approach (or to find the intuiton/motivation behind theory) you can check out link. For a more mathematical (measure-theoretic) approach have a look at say link.
            – Jean-Pierre de Villiers
            Nov 22 at 15:05


















          up vote
          0
          down vote













          The language $A$ dependent on $B$ is a bit vague, since it may involve something wherein there is a causal dependency that $A$ happens because $B$ does. Stochastic independence avoids that: Events $A,B$ are independent iff $P(A land B)=P(A)P(B).$ So $A,B$ are dependent iff $P(A land B)neq P(A)P(B),$ and the latter is symmetric in $A,B.$ I don't know what rules one uses in case "A depends on B" means something other than stochastic dependence.






          share|cite|improve this answer























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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            1
            down vote



            accepted










            Yes, two events are dependent if and only if they are not independent.
            Since, if $mu$ is a probability measure, then events A and B are independent if and only if
            begin{equation}
            mu(Acap B)=mu(A)cdotmu(B)
            end{equation}

            and dependent otherwise. Since multiplication is commutative you easily get exactly what you're looking for.



            An easy example is a double coin toss: intuitively the result of tossing the first coin should have no effect on the second and vice-versa.






            share|cite|improve this answer





















            • Thanks, any recommendation on becoming more familiar with this stuff? I'm currently taking a class at a University that is covering this content a bit too fast for my liking.
              – Kohler Fryer
              Nov 22 at 14:46










            • I'm a big fan of the MIT open courseware and would thus recommend taking a peek at that and the books they used. This also depends on the flavour of probility you are doing. For a more statistical/applied approach (or to find the intuiton/motivation behind theory) you can check out link. For a more mathematical (measure-theoretic) approach have a look at say link.
              – Jean-Pierre de Villiers
              Nov 22 at 15:05















            up vote
            1
            down vote



            accepted










            Yes, two events are dependent if and only if they are not independent.
            Since, if $mu$ is a probability measure, then events A and B are independent if and only if
            begin{equation}
            mu(Acap B)=mu(A)cdotmu(B)
            end{equation}

            and dependent otherwise. Since multiplication is commutative you easily get exactly what you're looking for.



            An easy example is a double coin toss: intuitively the result of tossing the first coin should have no effect on the second and vice-versa.






            share|cite|improve this answer





















            • Thanks, any recommendation on becoming more familiar with this stuff? I'm currently taking a class at a University that is covering this content a bit too fast for my liking.
              – Kohler Fryer
              Nov 22 at 14:46










            • I'm a big fan of the MIT open courseware and would thus recommend taking a peek at that and the books they used. This also depends on the flavour of probility you are doing. For a more statistical/applied approach (or to find the intuiton/motivation behind theory) you can check out link. For a more mathematical (measure-theoretic) approach have a look at say link.
              – Jean-Pierre de Villiers
              Nov 22 at 15:05













            up vote
            1
            down vote



            accepted







            up vote
            1
            down vote



            accepted






            Yes, two events are dependent if and only if they are not independent.
            Since, if $mu$ is a probability measure, then events A and B are independent if and only if
            begin{equation}
            mu(Acap B)=mu(A)cdotmu(B)
            end{equation}

            and dependent otherwise. Since multiplication is commutative you easily get exactly what you're looking for.



            An easy example is a double coin toss: intuitively the result of tossing the first coin should have no effect on the second and vice-versa.






            share|cite|improve this answer












            Yes, two events are dependent if and only if they are not independent.
            Since, if $mu$ is a probability measure, then events A and B are independent if and only if
            begin{equation}
            mu(Acap B)=mu(A)cdotmu(B)
            end{equation}

            and dependent otherwise. Since multiplication is commutative you easily get exactly what you're looking for.



            An easy example is a double coin toss: intuitively the result of tossing the first coin should have no effect on the second and vice-versa.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Nov 22 at 14:04









            Jean-Pierre de Villiers

            435




            435












            • Thanks, any recommendation on becoming more familiar with this stuff? I'm currently taking a class at a University that is covering this content a bit too fast for my liking.
              – Kohler Fryer
              Nov 22 at 14:46










            • I'm a big fan of the MIT open courseware and would thus recommend taking a peek at that and the books they used. This also depends on the flavour of probility you are doing. For a more statistical/applied approach (or to find the intuiton/motivation behind theory) you can check out link. For a more mathematical (measure-theoretic) approach have a look at say link.
              – Jean-Pierre de Villiers
              Nov 22 at 15:05


















            • Thanks, any recommendation on becoming more familiar with this stuff? I'm currently taking a class at a University that is covering this content a bit too fast for my liking.
              – Kohler Fryer
              Nov 22 at 14:46










            • I'm a big fan of the MIT open courseware and would thus recommend taking a peek at that and the books they used. This also depends on the flavour of probility you are doing. For a more statistical/applied approach (or to find the intuiton/motivation behind theory) you can check out link. For a more mathematical (measure-theoretic) approach have a look at say link.
              – Jean-Pierre de Villiers
              Nov 22 at 15:05
















            Thanks, any recommendation on becoming more familiar with this stuff? I'm currently taking a class at a University that is covering this content a bit too fast for my liking.
            – Kohler Fryer
            Nov 22 at 14:46




            Thanks, any recommendation on becoming more familiar with this stuff? I'm currently taking a class at a University that is covering this content a bit too fast for my liking.
            – Kohler Fryer
            Nov 22 at 14:46












            I'm a big fan of the MIT open courseware and would thus recommend taking a peek at that and the books they used. This also depends on the flavour of probility you are doing. For a more statistical/applied approach (or to find the intuiton/motivation behind theory) you can check out link. For a more mathematical (measure-theoretic) approach have a look at say link.
            – Jean-Pierre de Villiers
            Nov 22 at 15:05




            I'm a big fan of the MIT open courseware and would thus recommend taking a peek at that and the books they used. This also depends on the flavour of probility you are doing. For a more statistical/applied approach (or to find the intuiton/motivation behind theory) you can check out link. For a more mathematical (measure-theoretic) approach have a look at say link.
            – Jean-Pierre de Villiers
            Nov 22 at 15:05










            up vote
            0
            down vote













            The language $A$ dependent on $B$ is a bit vague, since it may involve something wherein there is a causal dependency that $A$ happens because $B$ does. Stochastic independence avoids that: Events $A,B$ are independent iff $P(A land B)=P(A)P(B).$ So $A,B$ are dependent iff $P(A land B)neq P(A)P(B),$ and the latter is symmetric in $A,B.$ I don't know what rules one uses in case "A depends on B" means something other than stochastic dependence.






            share|cite|improve this answer



























              up vote
              0
              down vote













              The language $A$ dependent on $B$ is a bit vague, since it may involve something wherein there is a causal dependency that $A$ happens because $B$ does. Stochastic independence avoids that: Events $A,B$ are independent iff $P(A land B)=P(A)P(B).$ So $A,B$ are dependent iff $P(A land B)neq P(A)P(B),$ and the latter is symmetric in $A,B.$ I don't know what rules one uses in case "A depends on B" means something other than stochastic dependence.






              share|cite|improve this answer

























                up vote
                0
                down vote










                up vote
                0
                down vote









                The language $A$ dependent on $B$ is a bit vague, since it may involve something wherein there is a causal dependency that $A$ happens because $B$ does. Stochastic independence avoids that: Events $A,B$ are independent iff $P(A land B)=P(A)P(B).$ So $A,B$ are dependent iff $P(A land B)neq P(A)P(B),$ and the latter is symmetric in $A,B.$ I don't know what rules one uses in case "A depends on B" means something other than stochastic dependence.






                share|cite|improve this answer














                The language $A$ dependent on $B$ is a bit vague, since it may involve something wherein there is a causal dependency that $A$ happens because $B$ does. Stochastic independence avoids that: Events $A,B$ are independent iff $P(A land B)=P(A)P(B).$ So $A,B$ are dependent iff $P(A land B)neq P(A)P(B),$ and the latter is symmetric in $A,B.$ I don't know what rules one uses in case "A depends on B" means something other than stochastic dependence.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Nov 22 at 14:07

























                answered Nov 22 at 14:02









                coffeemath

                1,9801413




                1,9801413






























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