document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200745>Question 1200745</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A court consists of 3 judges. Two of them, independently of each other, make correct decisions with a probability of p. The third judge agrees with the first two decisions when they match. In case of different opinions of the first two judges, the third judge decides on his own and makes a mistake with a probability of q. What is the probability that the court will not make a mistake if the verdict is decided by a majority vote? ([p, q] = [0.5, 0.43]) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #834954 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</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=834954\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red size=4>0.535</font><BR>\n" );
document.write( "This value is exact without any rounding done to it.\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "Let's give the judges code names: {J1,J2,J3}<BR>\n" );
document.write( "We'll have them be presented in the order that was mentioned in the instructions.\r<BR>\n" );
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document.write( "J1 and J2 make their decisions independently of each other. <BR>\n" );
document.write( "The probability they reach a correct decision is p = 0.5 for each of those two judges.<BR>\n" );
document.write( "If {J1,J2} reach the same outcome, then J3 rules the same way.\r<BR>\n" );
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document.write( "If {J1,J2} reach differing outcomes, then J3 thinks for her/himself to reach their own decision (making a mistake with probability q=0.43).<BR>\n" );
document.write( "For this situation J3 makes the correct decision with probability 1-q=1-0.43=0.57\r<BR>\n" );
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document.write( "Here is a way to represent a two-way table of outcomes for the first two judges.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >J2 right</TD><TD >J2 wrong</TD></tr><tr><TD >J1 right</TD><TD ></TD><TD ></TD></tr><tr><TD >J1 wrong</TD><TD ></TD><TD ></TD></tr>\n" );
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document.write( "For example, the upper left corner has both J1,J2 give the correct ruling.\r<BR>\n" );
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document.write( "Here is a non-tabular format to represent those four outcomes:<BR>\n" );
document.write( "J1 right, J2 right<BR>\n" );
document.write( "J1 right, J2 wrong<BR>\n" );
document.write( "J1 wrong, J2 right<BR>\n" );
document.write( "J1 wrong, J2 wrong\r<BR>\n" );
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document.write( "Let's list those four outcomes in a new table. <BR>\n" );
document.write( "We'll have them as the 4 rows. The 2 columns will be the outcomes for J3.<BR>\n" );
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document.write( "<tr><TD ></TD><TD >J3 right</TD><TD >J3 wrong</TD></tr><tr><TD >J1 right,J2 right</TD><TD >A</TD><TD ></TD></tr><tr><TD >J1 right,J2 wrong</TD><TD >B</TD><TD >C</TD></tr><tr><TD >J1 wrong,J2 right</TD><TD >D</TD><TD >E</TD></tr><tr><TD >J1 wrong,J2 wrong</TD><TD ></TD><TD >F</TD></tr>\n" );
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document.write( "I then added letters A through F to represent the various possible scenarios.<BR>\n" );
document.write( "Scenario A is when all three judges make the correct decision.<BR>\n" );
document.write( "The cell next to A is blank because J3 doesn't make the wrong decision when her/his two other colleagues make the correct decision (remember J3 mimics the other two judges if they rule the same way). This also explains why we have a blank cell in the bottom left corner.<BR>\n" );
document.write( "Scenario B is when J1 gets it right, J2 gets it wrong, J3 gets it right. <BR>\n" );
document.write( "Scenario C is when J1 gets it right, J2 gets it wrong, J3 gets it wrong. <BR>\n" );
document.write( "And so on.\r<BR>\n" );
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document.write( "The letters A, B, and D correspond to situations where the court gets the correct outcome. <BR>\n" );
document.write( "This is because there are at least 2 judges that get the correct ruling. <BR>\n" );
document.write( "A = all 3 judges get it right<BR>\n" );
document.write( "B = J1 and J3 get it right<BR>\n" );
document.write( "D = J2 and J3 get it right<BR>\n" );
document.write( "All other situations involve at least 2 judges getting the wrong verdict; hence the wrong final outcome.\r<BR>\n" );
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document.write( "To find the probability that the court gets the correct decision, we need to calculate the following: <BR>\n" );
document.write( "P(A)<BR>\n" );
document.write( "P(B)<BR>\n" );
document.write( "P(D)\r<BR>\n" );
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document.write( "P(A) = P(J1 right,J2 right)<BR>\n" );
document.write( "P(A) = P(J1 right)*P(J2 right)<BR>\n" );
document.write( "P(A) = p*p<BR>\n" );
document.write( "P(A) = p^2 <BR>\n" );
document.write( "P(A) = 0.5^2 <BR>\n" );
document.write( "P(A) = 0.25<BR>\n" );
document.write( "This works because J1,J2 are independent of each other.<BR>\n" );
document.write( "J3 does not alter the outcome here, so we ignore the probability associated with this judge.\r<BR>\n" );
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document.write( "P(B) = P(J1 right, J2 wrong, J3 right)<BR>\n" );
document.write( "P(B) = P(J1 right)*P(J2 wrong)*P(J3 right)<BR>\n" );
document.write( "P(B) = p*(1-p)*(1-q)<BR>\n" );
document.write( "P(B) = 0.5*(1-0.5)*(1-0.43)<BR>\n" );
document.write( "P(B) = 0.1425\r<BR>\n" );
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document.write( "P(D) = P(J1 wrong, J2 right, J3 right)<BR>\n" );
document.write( "P(D) = P(J1 wrong)*P(J2 right)*P(J3 right)<BR>\n" );
document.write( "P(D) = (1-p)*p*(1-q)<BR>\n" );
document.write( "P(D) = (1-0.5)*0.5*(1-0.43)<BR>\n" );
document.write( "P(D) = 0.1425\r<BR>\n" );
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document.write( "Conclusion:<BR>\n" );
document.write( "P(A) = 0.25<BR>\n" );
document.write( "P(B) = 0.1425<BR>\n" );
document.write( "P(D) = 0.1425\r<BR>\n" );
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document.write( "P(A)+P(B)+P(D) = 0.25+0.1425+0.1425 = <font color=red>0.535</font> represents the probability the court gets the right decision.<BR>\n" );
document.write( "The court gets it right exactly 53.5% of the time.\r<BR>\n" );
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document.write( "This is around 50%, so it's about as good as a coin toss in my opinion. Those don't seem like good odds. <BR>\n" );
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