document.write( "Question 1192753: Suppose that A and B are independent. Show that A' is independent of B' . \n" ); document.write( "

Algebra.Com's Answer #824647 by ikleyn(52776) $\"\"$ $\"About$

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\n" ); document.write( "Suppose that A and B are independent. Show that A' is independent of B' .
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document.write( "Let U be the universal set, to which A and B are the subsets.\r\n" );
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document.write( "Then  (A' ∩ B')  are those elements of the universal set U\r\n" );
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document.write( "that belong neither A nor B.  In other words,  (A' ∩ B') = U \ (A U B).\r\n" );
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document.write( "    Therefore,  P(A' ∩ B') = 1 - P(A U B).\r\n" );
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document.write( "From the other side,  P(A U B) = P(A) + P(B) - P(A ∩ B),  according to the basic formula\r\n" );
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document.write( "So we have\r\n" );
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document.write( "    P(A' ∩ B') = 1 - P(A U B) = 1 - P(A) - P(B) + P(A ∩ B) = (1-P(A))*(1-P(B)) = P(A')*P(B').\r\n" );
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document.write( "Thus we proved that  P(A' ∩ B') = P(A')*(P(B').\r\n" );
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document.write( "It means that events A' and B' are independent.\r\n" );
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