document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=320792>Question 320792</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A sock drawer contains 6 socks, 3 of which are definitely blue and the other 3 of which are either blue or black (exactly how many of each is unknown). If the probability of picking two blue socks in a given random selection of two socks is 2 /3, how many black socks must the drawer contain? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #229740 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=229740\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> represent the number of blue socks in the drawer before the first selection is made.  Then we know the probability of selecting a blue sock on the first trial is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{x}{6}\">.  Given that a blue sock is selected on the first trial, we know that there are one fewer total socks and one fewer blue socks, hence the probability of drawing a blue sock on the second trial is: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{x\ -\ 1}{5}\">.  Then the overall probability of drawing two blue socks is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(\frac{x}{6}\right)\left(\frac{x\,-\,1}{5}\right)\ =\ \frac{2}{3}\">\r<BR>\n" );
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document.write( "A little algebra gets us to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{x^2\ -\ x}{30}\ =\ \frac{20}{30}\">\r<BR>\n" );
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document.write( "Hence:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2\ -\ x\ -\ 20\ =\ 0\">\r<BR>\n" );
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document.write( "Which factors to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x\ +\ 4)(x\ -\ 5)\ =\ 0\">\r<BR>\n" );
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document.write( "Toss out the negative root (minus 4 blue socks?  Ludicrous!) and we see that we must have 5 blue socks out of 6 at the start in order to be only 2/3 certain of drawing a pair.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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