Total probability as the sum of conditional probabilities











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Suppose I have 2 events $C_1$ and $C_2$ so that $C_1$, $C_2$, and their complements ($C_1'$ and $C_2'$) partition the probability space. Let the event I'm interested in be denoted by $X$. $C_1$ and $C_2$ and their complements are conditionally independent given $X$. The probability of $X$ is given by,



$$P(X)=P(X,C_1,C_2)+P(X,C_1',C_2)+P(X,C_1,C_2')+P(X,C_1',C_2')$$
Using the product rule, this can be expressed as follow,
$$P(X|C_1)P(C_1)P(C_2|X,C_1)+P(X|C_1')P(C_1')P(C_2|X,C_1')+P(X|C_1)P(C_1)P(C_2'|X,C_1)+P(X|C_1)P(C_1')P(C_2'|X,C_1').$$
Secondly, using the conditional independence of $C_1$ and $C_2$ (and their complements) given $X$, this is,
$$P(X|C_1)P(C_1)P(C_2|X)+P(X|C_1')P(C_1')P(C_2|X)+P(X|C_1)P(C_1)P(C_2'|X)+P(X|C_1')P(C_1')P(C_2'|X)$$
Have I done this correctly? I'll accept your answer if you can say yes or no, and refer to the relevant rules I've invoked or failed to invoke.










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  • "$C_1,C_2,C_1', text{ and }C_2'$ partition the probability space..." Are thry disjoint? Or $C_j'=C_j^c$?
    – d.k.o.
    Nov 21 at 17:50












  • Yes, they are complements.
    – user617643
    Nov 21 at 18:37










  • @d.k.o. Any idea?
    – user617643
    Nov 21 at 22:58















up vote
1
down vote

favorite












Suppose I have 2 events $C_1$ and $C_2$ so that $C_1$, $C_2$, and their complements ($C_1'$ and $C_2'$) partition the probability space. Let the event I'm interested in be denoted by $X$. $C_1$ and $C_2$ and their complements are conditionally independent given $X$. The probability of $X$ is given by,



$$P(X)=P(X,C_1,C_2)+P(X,C_1',C_2)+P(X,C_1,C_2')+P(X,C_1',C_2')$$
Using the product rule, this can be expressed as follow,
$$P(X|C_1)P(C_1)P(C_2|X,C_1)+P(X|C_1')P(C_1')P(C_2|X,C_1')+P(X|C_1)P(C_1)P(C_2'|X,C_1)+P(X|C_1)P(C_1')P(C_2'|X,C_1').$$
Secondly, using the conditional independence of $C_1$ and $C_2$ (and their complements) given $X$, this is,
$$P(X|C_1)P(C_1)P(C_2|X)+P(X|C_1')P(C_1')P(C_2|X)+P(X|C_1)P(C_1)P(C_2'|X)+P(X|C_1')P(C_1')P(C_2'|X)$$
Have I done this correctly? I'll accept your answer if you can say yes or no, and refer to the relevant rules I've invoked or failed to invoke.










share|cite|improve this question









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user617643 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  • "$C_1,C_2,C_1', text{ and }C_2'$ partition the probability space..." Are thry disjoint? Or $C_j'=C_j^c$?
    – d.k.o.
    Nov 21 at 17:50












  • Yes, they are complements.
    – user617643
    Nov 21 at 18:37










  • @d.k.o. Any idea?
    – user617643
    Nov 21 at 22:58













up vote
1
down vote

favorite









up vote
1
down vote

favorite











Suppose I have 2 events $C_1$ and $C_2$ so that $C_1$, $C_2$, and their complements ($C_1'$ and $C_2'$) partition the probability space. Let the event I'm interested in be denoted by $X$. $C_1$ and $C_2$ and their complements are conditionally independent given $X$. The probability of $X$ is given by,



$$P(X)=P(X,C_1,C_2)+P(X,C_1',C_2)+P(X,C_1,C_2')+P(X,C_1',C_2')$$
Using the product rule, this can be expressed as follow,
$$P(X|C_1)P(C_1)P(C_2|X,C_1)+P(X|C_1')P(C_1')P(C_2|X,C_1')+P(X|C_1)P(C_1)P(C_2'|X,C_1)+P(X|C_1)P(C_1')P(C_2'|X,C_1').$$
Secondly, using the conditional independence of $C_1$ and $C_2$ (and their complements) given $X$, this is,
$$P(X|C_1)P(C_1)P(C_2|X)+P(X|C_1')P(C_1')P(C_2|X)+P(X|C_1)P(C_1)P(C_2'|X)+P(X|C_1')P(C_1')P(C_2'|X)$$
Have I done this correctly? I'll accept your answer if you can say yes or no, and refer to the relevant rules I've invoked or failed to invoke.










share|cite|improve this question









New contributor




user617643 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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Suppose I have 2 events $C_1$ and $C_2$ so that $C_1$, $C_2$, and their complements ($C_1'$ and $C_2'$) partition the probability space. Let the event I'm interested in be denoted by $X$. $C_1$ and $C_2$ and their complements are conditionally independent given $X$. The probability of $X$ is given by,



$$P(X)=P(X,C_1,C_2)+P(X,C_1',C_2)+P(X,C_1,C_2')+P(X,C_1',C_2')$$
Using the product rule, this can be expressed as follow,
$$P(X|C_1)P(C_1)P(C_2|X,C_1)+P(X|C_1')P(C_1')P(C_2|X,C_1')+P(X|C_1)P(C_1)P(C_2'|X,C_1)+P(X|C_1)P(C_1')P(C_2'|X,C_1').$$
Secondly, using the conditional independence of $C_1$ and $C_2$ (and their complements) given $X$, this is,
$$P(X|C_1)P(C_1)P(C_2|X)+P(X|C_1')P(C_1')P(C_2|X)+P(X|C_1)P(C_1)P(C_2'|X)+P(X|C_1')P(C_1')P(C_2'|X)$$
Have I done this correctly? I'll accept your answer if you can say yes or no, and refer to the relevant rules I've invoked or failed to invoke.







probability probability-theory






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edited Nov 22 at 0:18





















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asked Nov 21 at 17:31









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  • "$C_1,C_2,C_1', text{ and }C_2'$ partition the probability space..." Are thry disjoint? Or $C_j'=C_j^c$?
    – d.k.o.
    Nov 21 at 17:50












  • Yes, they are complements.
    – user617643
    Nov 21 at 18:37










  • @d.k.o. Any idea?
    – user617643
    Nov 21 at 22:58


















  • "$C_1,C_2,C_1', text{ and }C_2'$ partition the probability space..." Are thry disjoint? Or $C_j'=C_j^c$?
    – d.k.o.
    Nov 21 at 17:50












  • Yes, they are complements.
    – user617643
    Nov 21 at 18:37










  • @d.k.o. Any idea?
    – user617643
    Nov 21 at 22:58
















"$C_1,C_2,C_1', text{ and }C_2'$ partition the probability space..." Are thry disjoint? Or $C_j'=C_j^c$?
– d.k.o.
Nov 21 at 17:50






"$C_1,C_2,C_1', text{ and }C_2'$ partition the probability space..." Are thry disjoint? Or $C_j'=C_j^c$?
– d.k.o.
Nov 21 at 17:50














Yes, they are complements.
– user617643
Nov 21 at 18:37




Yes, they are complements.
– user617643
Nov 21 at 18:37












@d.k.o. Any idea?
– user617643
Nov 21 at 22:58




@d.k.o. Any idea?
– user617643
Nov 21 at 22:58










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Indeed, if you do have that conditional independence, then that will be okay.





PS: It is the non-empty pairwise intersections of the sets and their complements that partition the space.






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    1 Answer
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    active

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    up vote
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    accepted










    Indeed, if you do have that conditional independence, then that will be okay.





    PS: It is the non-empty pairwise intersections of the sets and their complements that partition the space.






    share|cite|improve this answer

























      up vote
      1
      down vote



      accepted










      Indeed, if you do have that conditional independence, then that will be okay.





      PS: It is the non-empty pairwise intersections of the sets and their complements that partition the space.






      share|cite|improve this answer























        up vote
        1
        down vote



        accepted







        up vote
        1
        down vote



        accepted






        Indeed, if you do have that conditional independence, then that will be okay.





        PS: It is the non-empty pairwise intersections of the sets and their complements that partition the space.






        share|cite|improve this answer












        Indeed, if you do have that conditional independence, then that will be okay.





        PS: It is the non-empty pairwise intersections of the sets and their complements that partition the space.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 22 at 0:18









        Graham Kemp

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