A fair die is rolled. Consider events E = {1,3,5}, F = {2,3} and G = {2,3,4,5}

Find


(i) P(E|F) and P(F|E)


(ii) P(E|G) and P(G|E)


(iii) P((E F)|G) and P ((E F)|G)


The sample space for the given experiment will be:

S = {1, 2, 3, 4, 5, 6}


Here, E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5} ……….(I)


……….(II)


Now, E F = {3}, F G = {2, 3}, E G = {3, 5} ……….(III)


……….(IV)


(i) We know that


By definition of conditional probability,



[Using (II) and (IV)]



Similarly, we have


[Using (II) and (IV)]



(ii) We know that


By definition of conditional probability,





Similarly, we have




(iii) Clearly, from (I), we have


E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5}


E F = {1, 2, 3, 5}


(E F) G = {2, 3, 5}



……….(V)


Now, we know that


By definition of conditional probability,



[Using (II) and (V)]



Similarly, we have E F = {3} [Using (III)]


And G = {2, 3, 4, 5} [Using (I)]


(E F) G = {3}


……….(VI)


So,


[Using (II) and (VI)]



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