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 #828557 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=828557\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "There are of course many ways to set up and solve this problem; and the standard methods taught in typical textbooks are not always the easiest way.<br><BR>\n" );
document.write( "The numbers in this problem lend themselves to a quick and easy solution using a non-standard method.<br><BR>\n" );
document.write( "We are given...<BR>\n" );
document.write( "(1) P(A) = P(B) = x  (Teams A and B have equal probability of winning)<BR>\n" );
document.write( "(2) P(D) = y; P(C) = 2y  (Team C is twice as likely to win as team D)<BR>\n" );
document.write( "(3) P(A)+P(C) = x+2y = 0.6  (The probability that either A or C wins is 0.6)<br><BR>\n" );
document.write( "Since the sum of all the probabilities is 1, we also have...<BR>\n" );
document.write( "(4) x+x+y+2y = 2x+3y = 1<br><BR>\n" );
document.write( "So we have two equations that we can solve to find the solution to the problem.  We could of course use the standard methods of substitution or elimination; but there is what I think is a much easier path to the answer.<br><BR>\n" );
document.write( "(4) tells us 2x+3y=1; (3) tells us x+2y=0.6.  Comparing those two gives us<br><BR>\n" );
document.write( "(5) x+y=0.4<br><BR>\n" );
document.write( "Then comparing (5) and (3) gives us y=0.2.<br><BR>\n" );
document.write( "And after that everything is easy:<BR>\n" );
document.write( "P(D) = y = 0.2<BR>\n" );
document.write( "P(C) = 2y = 0.4<BR>\n" );
document.write( "That leaves P(A)+P(B) = 0.4; and since P(A)=P(B), P(A)=P(B)=0.2.<br><BR>\n" );
document.write( "ANSWERS:<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<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );