document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192064>Question 1192064</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Three judges A, B and C must make a decision by majority's vote. They make their individual decision independently. It is known that the judges make a correct decision with probabilities A - 0.79, B - 0.65 and C 0.79. What is the chance that their decision will be correct? If their collective decision was correct, what is the chance of judge's A decision being the right one? Or what is the probability that only B and C had the right decision? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823963 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=823963\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "P(A) = probability that judge A makes the correct decision<BR>\n" );
document.write( "P(B) and P(C) represent similar ideas for the other two judges.\r<BR>\n" );
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document.write( "Given probabilities<BR>\n" );
document.write( "P(A) = 0.79<BR>\n" );
document.write( "P(B) = 0.65<BR>\n" );
document.write( "P(C) = 0.79\r<BR>\n" );
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document.write( "Their complements<BR>\n" );
document.write( "P(~A) = 1 - P(A) = 1 - 0.79 = 0.21<BR>\n" );
document.write( "P(~B) = 1 - P(B) = 1 - 0.65 = 0.35<BR>\n" );
document.write( "P(~C) = 1 - P(C) = 1 - 0.79 = 0.21<BR>\n" );
document.write( "which represent the probabilities of making the incorrect decision.\r<BR>\n" );
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document.write( "P(A and B only) = P(A, B, ~C)<BR>\n" );
document.write( "P(A and B only) = P(A)*P(B)*P(~C)<BR>\n" );
document.write( "P(A and B only) = 0.79*0.65*0.21<BR>\n" );
document.write( "P(A and B only) = 0.107835<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+0.107835 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+0.107835\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "P(B and C only) = P(~A, B, C)<BR>\n" );
document.write( "P(B and C only) = P(~A)*P(B)*P(C)<BR>\n" );
document.write( "P(B and C only) = 0.21*0.65*0.79<BR>\n" );
document.write( "P(B and C only) = 0.107835<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+0.107835 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+0.107835\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "P(A and C only) = P(A, ~B, C)<BR>\n" );
document.write( "P(A and C only) = P(A)*P(~B)*P(C)<BR>\n" );
document.write( "P(A and C only) = 0.79*0.35*0.79<BR>\n" );
document.write( "P(A and C only) = 0.218435<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B3%5D+=+0.218435 ALIGN=MIDDLE  ALT=\"x%5B3%5D+=+0.218435\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "P(A and B and C) = P(A)*P(B)*P(C)<BR>\n" );
document.write( "P(A and B and C) = 0.79*0.65*0.79<BR>\n" );
document.write( "P(A and B and C) = 0.405665<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B4%5D+=+0.405665 ALIGN=MIDDLE  ALT=\"x%5B4%5D+=+0.405665\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Cases <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D ALIGN=MIDDLE  ALT=\"x%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B3%5D ALIGN=MIDDLE  ALT=\"x%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > represent situations where exactly two judges get the right decision.<BR>\n" );
document.write( "Case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B4%5D ALIGN=MIDDLE  ALT=\"x%5B4%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is when all three judges make the correct ruling.<BR>\n" );
document.write( "All four represent when at least two judges get the correct ruling.\r<BR>\n" );
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document.write( "Add up the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D ALIGN=MIDDLE  ALT=\"x%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B4%5D ALIGN=MIDDLE  ALT=\"x%5B4%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D%2Bx%5B2%5D%2Bx%5B3%5D%2Bx%5B4%5D+=+0.107835%2B0.107835%2B0.218435%2B0.405665+=+0.83977 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The probability that at least two judges reach the correct decision, and therefore get the correct overall ruling, is 0.83977\r<BR>\n" );
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document.write( "Cases <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D ALIGN=MIDDLE  ALT=\"x%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B3%5D ALIGN=MIDDLE  ALT=\"x%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B4%5D ALIGN=MIDDLE  ALT=\"x%5B4%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > represent situations where judge A made the correct ruling that led to the overall ruling being correct.<BR>\n" );
document.write( "The sum of these x values is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D%2Bx%5B3%5D%2Bx%5B4%5D+=+0.731935 ALIGN=MIDDLE  ALT=\"x%5B1%5D%2Bx%5B3%5D%2Bx%5B4%5D+=+0.731935\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Dividing that second sum over the first sum calculated earlier will get us<BR>\n" );
document.write( "0.731935/0.83977 = 0.871590<BR>\n" );
document.write( "which is approximate.<BR>\n" );
document.write( "This is the probability of judge A being correct given the overall ruling was correct.<BR>\n" );
document.write( "This takes care of the second question mentioned.\r<BR>\n" );
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document.write( "For the third question, we go back to case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D ALIGN=MIDDLE  ALT=\"x%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is when judges B and C are correct, but judge A is not correct. \r<BR>\n" );
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document.write( "------------------------------------------------------------------------<BR>\n" );
document.write( "Summary:\r<BR>\n" );
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document.write( "What is the chance that their decision will be correct? <BR>\n" );
document.write( "<font color=red>0.83977</font>\r<BR>\n" );
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document.write( "If their collective decision was correct, what is the chance of judge's A decision being the right one? <BR>\n" );
document.write( "<font color=red>0.871590</font>\r<BR>\n" );
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document.write( "What is the probability that only B and C had the right decision? <BR>\n" );
document.write( "<font color=red>0.107835</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
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