document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1012911>Question 1012911</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a simple game, a six-sided die is rolled 4 times.  If all four rolls result in either 1 or 2, then the player wins $50 dollars.  If the player has to pay $10 to play the game, what are the expected winnings of the player over time if many games are played? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #629044 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=629044\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "If you roll a fair die once, the probability of getting a 1 or a 2 is 1/3.  The probability of 4 successes in 4 trials where the probability of success on one trial is:  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_4\left(4,\frac{1}{3}\right)\ =\ {{4}\choose{4}}\left(\frac{1}{3}\right)^4\left(1\,-\,\frac{1}{3}\right)^{4\,-\,4}\ =\ \left(\frac{1}{3}\right)^4\ =\ \frac{1}{81}\">\r<BR>\n" );
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document.write( "So over the long run you win 1 time every 81 times you play, and you lose 80 times every 81 times you play.  When you win, you pay $10 and get $50 back for a net gain of $40.  When you lose, you pay $10 and get nothing back for a net gain of -$10.\r<BR>\n" );
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document.write( "Hence your per game net is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{81}\,\times\,$50\ +\ \frac{80}{81}\,\times\,(-$10)\ \approx\ -$9.3827\">\r<BR>\n" );
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document.write( "And if you played 810 games, you should expect to lose about $7600 overall.\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" );
document.write( "My calculator said it, I believe it, that settles it\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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