Question

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

Solution

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

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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 ∪ F) | G) and P((E ∩ F) | G)

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