document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=441203>Question 441203</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 3.18 Suppose A and B are events with 0 < P(A) < 1 and 0 < P(B) < 1.<BR>\n" );
document.write( "a. If A and B are disjoint, can they be independent?<BR>\n" );
document.write( "b. If A and B are independent, can they be disjoint?<BR>\n" );
document.write( "c. If A &#8834; B, can A and B be independent?<BR>\n" );
document.write( "d. If A and B are independent, can A and A &#8746; B be independent </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #304541 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=304541\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a.  A, B disjoint means P(A and B) = 0.  Since P(A), P(B) > 0, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28A%29%2AP%28B%29+%3E+0 ALIGN=MIDDLE  ALT=\"P%28A%29%2AP%28B%29+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, hence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28A%29%2AP%28B%29+%3C%3E+P%28A+and+B%29 ALIGN=MIDDLE  ALT=\"P%28A%29%2AP%28B%29+%3C%3E+P%28A+and+B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and so are not independent.\r<BR>\n" );
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document.write( "b.  A, B independent means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28A%29%2AP%28B%29+=+P%28A+and+B%29 ALIGN=MIDDLE  ALT=\"P%28A%29%2AP%28B%29+=+P%28A+and+B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28A%29+%2B+P%28B%29+=+P%28A%29+%2B+P%28B%29+-+P%28A+and+B%29+=+P%28A%29+%2B+P%28B%29+-+P%28A%29%2AP%28B%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Unless P(A)*P(B)= 0, A and B can never be disjoint.  But from the given, P(A)*P(B) > 0, hence A, B are never disjoint.\r<BR>\n" );
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document.write( "c. If A &#8834; B, then P(A and B) = P(A).  If we assume independence, then <BR>\n" );
document.write( "P(A) = P(A)*P(B), which means <BR>\n" );
document.write( "P(B) = 1, a contradiction of the hypothesis that 0 < P(B) < 1.  Hence A, B are not independent.\r<BR>\n" );
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document.write( "d. P(A and (A &#8746; B)) = P(A), since A &#8834; A U B.  Hence the argument is reduced to one like in part (c) above, the result being the same. </SPAN>\n" );
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