document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=225723>Question 225723</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I posted this problem earlier but no one responded. I have figured out every problem except this one and need some help. particularly with how to formulate:\r<BR>\n" );
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document.write( "\"A fair die is rolled nine times. What is the probability that an odd number 1,3, or 5 will occur LESS than three times?\" \r<BR>\n" );
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document.write( "If there are six sides to a fair dice, I would start like this 3/6 ^9 or 1/2^9 which would equal 1/512...I am lost from there. HELP!! Thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #168342 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=168342\">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 exactly <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 is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_n(k)\ =\ \left(n\cr k\right)p^kq^{n-k}\">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\"> is the probability of success on one trial, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large q\"> the probability of failure on one trial and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(n\cr k\right)\ =\ \frac{n!}{k!(n-k)!}\">\r<BR>\n" );
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document.write( "For your problem you need to know the probability that an odd number occurs zero times or one time or two times.  So you will need the following sum:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_9(k<3)\ =\ \left(P_9(0)\right)\ +\ \left(P_9(1)\right)\ +\ \left(P_9(2)\right)\">\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\ =\ q\ =\ \frac{1}{2}\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p^kq^{n-k}\ =\ \left(\frac{1}{2}\right)^3\left(\frac{1}{2}\right)^6\ =\ \left(\frac{1}{2}\right)^9\ =\ \frac{1}{512}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(9\cr0\right)\ =\ \frac{9!}{0!(9-0)!}\ =\ 1\">\r<BR>\n" );
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document.write( "(Remember 0! = 1)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(9\cr1\right)\ =\ \frac{9!}{1!(9-1)!}\ =\ 9\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(9\cr2\right)\ =\ \frac{9!}{2!(9-2)!}\ =\ \frac{9!}{2!(7)!}\ =\ \frac{9\cdot8}{2}\ =\ 36\">\r<BR>\n" );
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document.write( "So:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_9(k<3)\ =\ \left(\frac{1}{512}\right)\ +\ \left(\frac{9}{512\right)\ +\ \left(\frac{36}{512}\right)\ =\ \frac{46}{512}\ =\ \frac{23}{256}\">\r<BR>\n" );
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document.write( "Which is to say that there is better than a 90% chance that it WON'T happen.\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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