document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1042698>Question 1042698</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider a biased coin with probability p=1/3 of landing heads. Suppose the coin is flipped 'n' times. Use the Chernoff bound to determine the smallest value for 'n' so that the probability that more than half of the coin flips come out heads is less than 0.001.<BR>\n" );
document.write( "a) 9 <BR>\n" );
document.write( "b) 249 <BR>\n" );
document.write( "c) 99 <BR>\n" );
document.write( "d) 499 <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #657753 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=657753\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You seem to be using the version <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X+%3E+%281%2Bepsilon%29%2Amu%29+%3C=+exp%28-epsilon%5E2%2F3%2Amu%29 ALIGN=MIDDLE  ALT=\"P%28X+%3E+%281%2Bepsilon%29%2Amu%29+%3C=+exp%28-epsilon%5E2%2F3%2Amu%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the Chernoff bound, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X+=+sum%28X%5Bi%5D%2C+i+=+1%2Cn%29 ALIGN=MIDDLE  ALT=\"X+=+sum%28X%5Bi%5D%2C+i+=+1%2Cn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and each <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%5Bi%5D ALIGN=MIDDLE  ALT=\"X%5Bi%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is Bernoulli.\r<BR>\n" );
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document.write( "You want <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X+%3E+n%2F2%29+%3C=+0.001 ALIGN=MIDDLE  ALT=\"P%28X+%3E+n%2F2%29+%3C=+0.001\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bepsilon%29mu+=+n%2F2 ALIGN=MIDDLE  ALT=\"%281%2Bepsilon%29mu+=+n%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.   Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu+=+n%2F3 ALIGN=MIDDLE  ALT=\"mu+=+n%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bepsilon%29%28n%2F3%29+=+n%2F2 ALIGN=MIDDLE  ALT=\"%281%2Bepsilon%29%28n%2F3%29+=+n%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=epsilon+=+1%2F2 ALIGN=MIDDLE  ALT=\"epsilon+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Next, let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=exp%28-epsilon%5E2%2F3%2Amu%29+=+exp%28-%281%2F2%29%5E2%2F3%2A%28n%2F3%29%29+=+e%5E%28-n%2F36%29+=+0.001 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-n%2F36+=+ln0.001 ALIGN=MIDDLE  ALT=\"-n%2F36+=+ln0.001\"   DISABLED_style= \"max-width:90%; height:auto;\" >  ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=-36%2Aln0.001++=+248.68 ALIGN=MIDDLE  ALT=\"n+=-36%2Aln0.001++=+248.68\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28n+=+249%29 ALIGN=MIDDLE  ALT=\"highlight%28n+=+249%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, rounded to the nearest whole number. </SPAN>\n" );
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