document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=852124>Question 852124</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the probability of a tennis player serving an ace is 1/6, what's the probability that the player will serve 3 aces out of 8 serves?\r<BR>\n" );
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document.write( "I know it's independent probability. I did it like this:<BR>\n" );
document.write( "for \"non-ace\" results, the prob. is 5/6<BR>\n" );
document.write( "for 8 serves, the prob of 3 aces is:\r<BR>\n" );
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document.write( "1/6 * 1/6 * 1/6 * 5/6 * 5/6 * 5/6 * 5/6 * 5/6 = 3125/1,679,616 = approx. 0.00186, or 0.186%, or 186/1000.\r<BR>\n" );
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document.write( "Am I even close? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #513162 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=513162\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Yes and no. <BR>\n" );
document.write( "I think you've got the concepts but you're missing a lot of terms. <BR>\n" );
document.write( "Since you don't know when the ace is served, the example you gave is only one possible outcome. <BR>\n" );
document.write( "It could have been <BR>\n" );
document.write( "AAANNNNN<BR>\n" );
document.write( "or <BR>\n" );
document.write( "ANAANNNN<BR>\n" );
document.write( "or<BR>\n" );
document.write( "NNNNNAAA<BR>\n" );
document.write( "So you need to capture all of the possible outcomes that each individually have a probability of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3125%2F6%5E8 ALIGN=MIDDLE  ALT=\"3125%2F6%5E8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can use Pascal's triangle with 9 terms (1,8,28,56,70,56,28,8,1).<BR>\n" );
document.write( "So for no aces, there is 1 term (NNNNNNNN)<BR>\n" );
document.write( "For 1 ace, 8 terms (ANNNNNNN, NANNNNNN, .... , NNNNNNNA)<BR>\n" );
document.write( "For 2 aces, 28 terms (too many terms to write out)<BR>\n" );
document.write( "For 3 aces, 56 terms.<BR>\n" );
document.write( "So then,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28P%283%29=56%2A%281%2F6%29%5E3%2A%285%2F6%29%5E5=0.10419%29 ALIGN=MIDDLE  ALT=\"highlight%28P%283%29=56%2A%281%2F6%29%5E3%2A%285%2F6%29%5E5=0.10419%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The other terms are,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%29=1%2A%281%2F6%29%5E0%2A%285%2F6%29%5E8=0.23257 ALIGN=MIDDLE  ALT=\"P%280%29=1%2A%281%2F6%29%5E0%2A%285%2F6%29%5E8=0.23257\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%281%29=8%2A%281%2F6%29%5E1%2A%285%2F6%29%5E7=0.37211 ALIGN=MIDDLE  ALT=\"P%281%29=8%2A%281%2F6%29%5E1%2A%285%2F6%29%5E7=0.37211\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%282%29=28%2A%281%2F6%29%5E2%2A%285%2F6%29%5E6=0.26048 ALIGN=MIDDLE  ALT=\"P%282%29=28%2A%281%2F6%29%5E2%2A%285%2F6%29%5E6=0.26048\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%284%29=70%2A%281%2F6%29%5E4%2A%285%2F6%29%5E4=0.02605 ALIGN=MIDDLE  ALT=\"P%284%29=70%2A%281%2F6%29%5E4%2A%285%2F6%29%5E4=0.02605\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%285%29=56%2A%281%2F6%29%5E5%2A%285%2F6%29%5E3=0.00417 ALIGN=MIDDLE  ALT=\"P%285%29=56%2A%281%2F6%29%5E5%2A%285%2F6%29%5E3=0.00417\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%286%29=28%2A%281%2F6%29%5E6%2A%285%2F6%29%5E2=0.00042 ALIGN=MIDDLE  ALT=\"P%286%29=28%2A%281%2F6%29%5E6%2A%285%2F6%29%5E2=0.00042\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%287%29=8%2A%281%2F6%29%5E7%2A%285%2F6%29%5E1+=0.000024 ALIGN=MIDDLE  ALT=\"P%287%29=8%2A%281%2F6%29%5E7%2A%285%2F6%29%5E1+=0.000024\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%288%29=1%2A%281%2F6%29%5E8%2A%285%2F6%29%5E0+=0.0000006 ALIGN=MIDDLE  ALT=\"P%288%29=1%2A%281%2F6%29%5E8%2A%285%2F6%29%5E0+=0.0000006\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If you add all the terms, they do sum to 1. </SPAN>\n" );
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