document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=325274>Question 325274</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You are given 12 to 1 odds against tossing 3 tails in 3 tosses of a fair coin. This means that you win $12 if you succeed and  you lose $1 if you fail. Find the expected value(to you) of the game. Round to the nearest cent. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #232946 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=232946\">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( "The probability of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> successes in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> trials where the probability of success on any given trial is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\"> is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE\ \ \ \ \ \ \ \ \ \ P_n(k,p)\ =\ \left(n\cr{k}\right)\left(p\right)^k\left(1\ -\ p\right)^{n\,-\,k}\">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(n\cr{k}\right)\"> is the number of combinations of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> things taken <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> at a time and is computed using <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{n!}{k!(n\,-\,k)!\">\r<BR>\n" );
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document.write( "For this problem, given a fair coin, the probability of success on any given trial is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{1}{2}\">, and you want 3 successes out of 3 trials, hence:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE\ \ \ \ \ \ \ \ \ \ P_3\left(3,\frac{1}{2}\right)\ =\ \left(3\cr{3}\right)\left(\frac{1}{2}\right)^3\left(\frac{1}{2}\right)^{0}\">\r<BR>\n" );
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document.write( "You can calculate this for yourself.  Hint:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(n\cr{n}\right)\ =\ 1\"> and anything to the zero power is 1.\r<BR>\n" );
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document.write( "That gives you the probability of winning.  The probability of losing is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1\ -\ p_w\">.\r<BR>\n" );
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document.write( "The expected outcome is the amount you lose whenever you lose times the probability of losing plus the amount you win whenever you win times the probability of winning.\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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