document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1045311>Question 1045311</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a duel between X and Y both duelists have 2 bullets. The first one to shoot is X, with a chance to hit of 30%, afterwards if X missed Y shoots with a chance to hit of 60%, afterwards if Y missed X shoots with a chance to hit of 40% and finally if X missed Y shoots with a chance to hit of 50%. What is the probability that Y will hit X? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #660767 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=660767\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We go by the assumption that once a duelist hits his opponent, then the hit opponent cannot hit back anymore, and the duel ends.\r<BR>\n" );
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document.write( "With that said, there are only 5 possible elements to this sample space, which are the following mutually exclusive events:\r<BR>\n" );
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document.write( "{X hit},<BR>\n" );
document.write( "{X no-hit, Y hit},<BR>\n" );
document.write( "{X no-hit, Y no-hit, X hit}<BR>\n" );
document.write( "{X no-hit, Y no-hit, X no-hit, Y hit}<BR>\n" );
document.write( "{X no-hit, Y no-hit, X no-hit, Y no-hit}.\r<BR>\n" );
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document.write( "This listing can also be illustrated using a TREE DIAGRAM, with the proper labeling of probabilities being indicated at every respective <BR>\n" );
document.write( "branch.\r<BR>\n" );
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document.write( "We are interested in cases 2 and 4 above, where Y ends up hitting X.\r<BR>\n" );
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document.write( "===> probability is 0.7*0.6 + 0.7*0.4*0.6*0.5 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%280.504%29 ALIGN=MIDDLE  ALT=\"highlight%280.504%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "N.B.: \r<BR>\n" );
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document.write( "The probability that X hits Y is done similarly (represented by cases 1 and 3), and is \r<BR>\n" );
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document.write( "0.3 + 0.7*0.4*0.4 = 0.412.\r<BR>\n" );
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document.write( "The probability that neither of them hits the other during the whole process is 0.7*0.4*0.6*0.5 = 0.084. (Both of them walk off unscathed.)\r<BR>\n" );
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document.write( "Checking the probabilities for all cases gives 0.504 + 0.412 + 0.084 = 1. </SPAN>\n" );
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