document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=576945>Question 576945</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am trying to calculate the odds similar to lottery with different conditions and don't know how to do it.  I am not trying to win the lottery.  It is for a board game variation similar to Bingo.\r<BR>\n" );
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document.write( "The conditions are:  45 numbers, pick 5, 15 numbers randomly chosen.  How do I calculate the odds of getting all 5 numbers being drawn?  Would I continue with setting up the as 1/45, 1/44, 1/43, 1/42, 1/41 and then 1/40?  or would I go 3/45, 3/44....?  Would I still use 120 as the dividing factor, or would I divide by 360?  Help would be appreciate.  Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #370029 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=370029\">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( "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 <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\"> is the probability of success on any given trial 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 calculated by <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( "You want 5 successes out of 5 trials where the probability of success on any given trial is 15/45 = 1/3, so the odds are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_5\left(5,\frac{1}{3}\right)\ =\ \left(5\cr 5\right\)\left(\frac{1}{3}\right)^5\left(\frac{2}{3}\right)^0\">\r<BR>\n" );
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document.write( "But since \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(n\cr n\right\)\ =\ 1\ \forall\,n\,\geq\,0\,\in\,\mathbb{Z}\">\r<BR>\n" );
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document.write( "and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a^0\ =\ 1\ \forall\,a\,\in\,\mathbb{R}\">\r<BR>\n" );
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document.write( "this reduces to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_5\left(5,\frac{1}{3}\right)\ =\ \left(\frac{1}{3}\right)^5\ =\ \frac{1}{243}\">\r<BR>\n" );
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document.write( "Which is the probability, but you asked for the odds.  Odds in favor are the probability of success divided by the probability of failure (which is 1 minus the probability of success), so the odds are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{P_5\left(5,\frac{1}{3}\right)}{1\ -\ P_5\left(5,\frac{1}{3}\right)}\ =\ \frac{\ \frac{1}{243}\ }{\ \frac{242}{243}\ }\ =\ \frac{1}{242}\">\r<BR>\n" );
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document.write( "1 in 242, or 242 to 1 against.\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<BR>\n" );
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