document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195929>Question 1195929</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Four teams A, B, C, and D compete in a tournament, and exactly one of them will<BR>\n" );
document.write( "win the tournament. Teams A and B have the same chance of winning the tournament. Team C is<BR>\n" );
document.write( "twice as likely to win the tournament as team D. The probability that either team A or team C wins<BR>\n" );
document.write( "the tournament is 0.6. Find the probabilities of each team winning the tournament. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #828551 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=828551\">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) = x<BR>\n" );
document.write( "P(B) = x<BR>\n" );
document.write( "P(C) = 2y<BR>\n" );
document.write( "P(D) = y<BR>\n" );
document.write( "These are the probabilities of each team winning the tournament<BR>\n" );
document.write( "The values are such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+x+%3C=+1 ALIGN=MIDDLE  ALT=\"0+%3C=+x+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+y+%3C=+1 ALIGN=MIDDLE  ALT=\"0+%3C=+y+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Those probabilities add to 1 since they represent 100% of all possible cases.<BR>\n" );
document.write( "P(A)+P(B)+P(C)+P(D) = 1<BR>\n" );
document.write( "x+x+2y+y = 1<BR>\n" );
document.write( "2x+3y = 1\r<BR>\n" );
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document.write( "The probability of team A or C winning is 0.6, so,<BR>\n" );
document.write( "P(A or C) = P(A) + P(C) ...  mutually exclusive<BR>\n" );
document.write( "P(A or C) = 0.6<BR>\n" );
document.write( "P(A) + P(C) = 0.6<BR>\n" );
document.write( "x+2y = 0.6<BR>\n" );
document.write( "x = -2y+0.6\r<BR>\n" );
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document.write( "Apply substitution<BR>\n" );
document.write( "2x+3y = 1<BR>\n" );
document.write( "2( x )+3y = 1<BR>\n" );
document.write( "2( -2y+0.6 )+3y = 1<BR>\n" );
document.write( "-4y+1.2+3y = 1<BR>\n" );
document.write( "-y+1.2 = 1<BR>\n" );
document.write( "-y = 1-1.2<BR>\n" );
document.write( "-y = -0.2<BR>\n" );
document.write( "y = 0.2<BR>\n" );
document.write( "Therefore, P(C) = 2*y = 2*0.2 = 0.4 and P(D) = y = 0.2\r<BR>\n" );
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document.write( "And,<BR>\n" );
document.write( "x = -2y+0.6<BR>\n" );
document.write( "x = -2(0.2)+0.6<BR>\n" );
document.write( "x = -0.4+0.6<BR>\n" );
document.write( "x = 0.2<BR>\n" );
document.write( "which means P(A) = P(B) = x = 0.2\r<BR>\n" );
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document.write( "<font color=red>Summary:</font><BR>\n" );
document.write( "P(A) = 0.2<BR>\n" );
document.write( "P(B) = 0.2<BR>\n" );
document.write( "P(C) = 0.4<BR>\n" );
document.write( "P(D) = 0.2\r<BR>\n" );
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document.write( "Check:<BR>\n" );
document.write( "P(A)+P(B)+P(C)+P(D) = 1<BR>\n" );
document.write( "0.2+0.2+0.4+0.2 = 1<BR>\n" );
document.write( "1 = 1<BR>\n" );
document.write( "and,<BR>\n" );
document.write( "P(A or C) = P(A) + P(C) = 0.2 + 0.4 = 0.6<BR>\n" );
document.write( "The answers are confirmed.<BR>\n" );
document.write( "</font> </SPAN>\n" );
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