document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1128416>Question 1128416</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A and B are two events.\r<BR>\n" );
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document.write( "Let P(A)=0.3 ,  P(B)=0.8 , and P(A&#8201;and B)=0.24 .\r<BR>\n" );
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document.write( "Which statement is true?\r<BR>\n" );
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document.write( "A and B are not independent events because P(A|B)=P(A) and P(B|A)=P(B) .\r<BR>\n" );
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document.write( "A and B are not independent events because  P(A|B)&#8800;P(A) .\r<BR>\n" );
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document.write( "A and B are independent events because P(A|B)=P(A) and P(B|A)=P(B) .\r<BR>\n" );
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document.write( "A and B are not independent events because P(A|B)=P(B) and  P(B|A)=P(A) . </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #744978 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=744978\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><BR>\n" );
document.write( "P(A) = 0.3<BR>\n" );
document.write( "P(B) = 0.8<BR>\n" );
document.write( "P(A and B) = 0.24\r<BR>\n" );
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document.write( "P(A | B) = P(A and B)/P(B)<BR>\n" );
document.write( "P(A | B) = 0.24/0.8<BR>\n" );
document.write( "P(A | B) = 0.3<BR>\n" );
document.write( "P(A | B) = P(A) <--- we'll use this fact\r<BR>\n" );
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document.write( "P(B | A) = P(B and A)/P(A)<BR>\n" );
document.write( "P(B | A) = P(A and B)/P(A)<BR>\n" );
document.write( "P(B | A) = 0.24/0.3<BR>\n" );
document.write( "P(B | A) = 0.8<BR>\n" );
document.write( "P(B | A) = P(B) <---- and this fact too\r<BR>\n" );
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document.write( "Since both <font color=red>P(A | B) = P(A) and P(B | A) = P(B)</font> are true, this means that <font color=red>A and B are independent events</font>.\r<BR>\n" );
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document.write( "Put another way: if we know that event A occurs, then that does not change the probability P(B). So P(B | A) = P(B) meaning B is independent of A. Similarly, P(A | B) = P(A) tells us that prior knowledge of event B happening does not change the probability P(A), therefore A is independent of B. \r<BR>\n" );
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document.write( "If either P(A|B) = P(A) or P(B|A) = P(B) were false equations, then A and B would be dependent events. <BR>\n" );
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