Question

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

Solution

Since fair die is rolled
$therefore space space space straight S space equals space open curly brackets 1 comma space 2 comma space 3 comma space 4 comma space 5 comma space 6 close curly brackets.$
   $straight E space equals open curly brackets 1 comma space 3 comma space 5 close curly brackets comma space space space straight F space equals space open curly brackets 2 comma space 3 comma close curly brackets comma space space straight G space equals space open curly brackets 2 comma space 3 comma space 4 comma space 5 close curly brackets$
$straight E intersection straight G space equals space open curly brackets 3 comma space 5 close curly brackets therefore space space space straight P left parenthesis straight E right parenthesis space equals space 3 over 6 comma space space straight P left parenthesis straight G right parenthesis space equals space 4 over 6 comma space space straight P left parenthesis straight E space intersection space straight G right parenthesis space equals 2 over 6 therefore space space space space space straight P left parenthesis straight E space left enclose straight G right parenthesis end enclose space equals space fraction numerator straight P left parenthesis straight E intersection straight G right parenthesis over denominator straight P left parenthesis straight G right parenthesis end fraction space equals space fraction numerator begin display style 2 over 6 end style over denominator begin display style 4 over 6 end style end fraction equals space 2 over 4 space equals space 1 half space space space space space space space space space space space space space space space straight P left parenthesis straight G space left enclose space straight E right parenthesis end enclose space equals space fraction numerator straight P left parenthesis straight G intersection straight E right parenthesis over denominator straight P left parenthesis straight E right parenthesis end fraction space equals space fraction numerator straight P left parenthesis straight E intersection straight G right parenthesis over denominator straight P left parenthesis straight E right parenthesis end fraction space equals space fraction numerator begin display style 2 over 6 end style over denominator begin display style 3 over 6 end style end fraction space equals space 2 over 3 space space space space space$

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Probability

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

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