document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=319371>Question 319371</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the probability of getting between 3 and 6 heads in 10 tosses of a fair coin? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #228689 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=228689\">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( "\"between 3 and 6\" is ambiguous.  Do you mean inclusive or exclusive of 3 and 6?  No matter, I'll show you what to do either way.\r<BR>\n" );
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document.write( "If you mean <i>inclusive</i> of 3 and 6, then the probability you seek is the probability of getting <i>exactly</i> 3 heads out of 10 tosses, plus the probability of getting exactly 4, plus exactly 5, plus exactly 6.\r<BR>\n" );
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document.write( "On the other hand, if you mean <i>exclusive</i>, then the probability you seek is the probability of exactly 4 heads out of 10 tosses plus the probability of exactly 5 heads out of 10 tosses.\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 out of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> trials where the probability of success on any individual 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)\ =\ \left(n\cr k\right\)p^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 has the value <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 a fair coin toss, the probability of a head on any given trial is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{1}{2}\">\r<BR>\n" );
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document.write( "So, for the inclusive case, you need to calculate the following:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_{10}(3)\ =\ \left(10\cr \,3\right\)\left(\frac{1}{2}\right)^3\left(\frac{1}{2}\right)^{7}\ =\ \left(10\cr \,3\right\)\left(\frac{1}{2}\right)^{10}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_{10}(4)\ =\ \left(10\cr \,4\right\)\left(\frac{1}{2}\right)^{10}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_{10}(5)\ =\ \left(10\cr \,5\right\)\left(\frac{1}{2}\right)^{10}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_{10}(6)\ =\ \left(10\cr \,6\right\)\left(\frac{1}{2}\right)^{10}\">\r<BR>\n" );
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document.write( "And sum the four results.\r<BR>\n" );
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document.write( "For the exclusive case, you only need to compute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_{10}(4)\ =\ \left(10\cr \,4\right\)\left(\frac{1}{2}\right)^{10}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_{10}(5)\ =\ \left(10\cr \,5\right\)\left(\frac{1}{2}\right)^{10}\">\r<BR>\n" );
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document.write( "and sum the two results\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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