Let E and F be events with $straight P left parenthesis straight E right parenthesis space equals space 3 over 5 comma space straight P left parenthesis straight F right parenthesis space equals space 3 over 10 space and space straight P left parenthesis straight E space intersection space straight F right parenthesis space equals space 1 fifth.$ Are E and F independent?

Solution

$straight P left parenthesis straight E right parenthesis space equals 3 over 5 comma space space space straight P left parenthesis straight F right parenthesis space equals 3 over 10 comma space space straight P left parenthesis straight E intersection straight F right parenthesis space equals 1 fifth$
$Now comma space straight P left parenthesis straight E right parenthesis. space straight P left parenthesis straight F right parenthesis space equals space 3 over 5 cross times 3 over 10 space equals 9 over 50 not equal to space straight P left parenthesis straight E intersection straight F right parenthesis space space space space space space space space space space open square brackets because space space space straight P left parenthesis straight E intersection straight F right parenthesis space equals space 1 fifth close square brackets therefore space space space space space space space space straight P left parenthesis straight E right parenthesis. space straight P left parenthesis straight F right parenthesis space space space not equal to space straight P left parenthesis straight E intersection straight F right parenthesis$

∴ E and F are not independent.

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