document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=761278>Question 761278</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A survey finds customers overpay for 4 out of every 10 items. Suppose a customer purchases 15 items.\r<BR>\n" );
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document.write( "a) the probability the customer overpays on none of the items is______? (round decimal to the nearest thousandth).<BR>\n" );
document.write( "b) The probability the customer overpays on at least 3 items is______? (round decimal to the nearest thousandth).\r<BR>\n" );
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document.write( "-I got .104 for the first answer is that correct? I can't figure out the second one! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #463163 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=463163\">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( "No.\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)\ =\ {{n}\choose{k}}\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 {{n}\choose{k}}\"> 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( "For part a you need <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P_{15}(0,0.4)\ =\ {{15}\choose{0}}\left(0.4\right)^0\left(0.6\right)^{15}\">\r<BR>\n" );
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document.write( "I'll let you do your own arithmetic.  (hint: n choose 0 = 1)\r<BR>\n" );
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document.write( "For part b, the straight forward approach is to calculate the above for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\ =\ 3\">, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\ =\ 4\">, then 5, 6, ... and so on up to 15. And then you have to add up all of your answers for these results.  Lots of VERY nasty arithmetic.  Looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_n(\geq k,p)\ =\ \sum_{i\,=\,k}^n\,{{n}\choose{i}}\left(p\right)^i\left(1\,-\,p\right)^{n\,-\,i}\">\r<BR>\n" );
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document.write( "Better approach:  The probability of AT LEAST 3 is equal to 1 minus the probability of LESS THAN 3.  The probability of less than 3 is the probability of zero (which is what you just calculated in part a) plus the probability of 1 plus the probability of 2.  Don't forget to subtract the sum from 1.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_n(<k,p)\ =\ 1\ -\ \sum_{i\,=\,0}^{k\,-\,1}\,{{n}\choose{i}}\left(p\right)^i\left(1\,-\,p\right)^{n\,-\,i}\">\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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