document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=239414>Question 239414</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Assume the likelihood that any flight on Northwest Airlines arrives within 15 minutes of the scheduled time is .90. We select four flights from yesterday for study.<BR>\n" );
document.write( "1.	What is the likelihood all four of the selected flights arrived within 15 minutes of the scheduled time? <BR>\n" );
document.write( "2.	What is the likelihood that none of the selected flights arrived within 15 minutes of the scheduled time?<BR>\n" );
document.write( "3.	What is the likelihood at least one of the selected flights did not arrive within 15 minutes of the scheduled time?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #175755 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=175755\">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 out of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> trials 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\,q^{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 a given trial, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large q\"> is the probability of failure on a given trial, and <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, or <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( "1.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_4(4)\ =\ \left(4\cr 4\right)\,(0.9)^4\,(0.1)^0\">\r<BR>\n" );
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document.write( "2.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_4(0)\ =\ \left(4\cr 0\right)\,(0.9)^0\,(0.1)^4\">\r<BR>\n" );
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document.write( "You get to do your own arithmetic.  Hints:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(n\cr n\right)\ =\ \left(n\cr 0\right)\ =\ 1\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a^0\ = 1\">\r<BR>\n" );
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document.write( "3.  At least one does NOT arrive within 15 minutes.  That is the probability that none, 1, 2, or 3 do not arrive within 15 minutes -- which is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_4(0)\ +\ P_4(1)\ +\ P_4(2)\ +\ P_4(3)\">.  That looks like a lot of calculation, but not really.  Since we know that either none or at least one must be late, then we can be certain that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_4(0)\ +\ P_4(1)\ +\ P_4(2)\ +\ P_4(3)\ +\ P_4(4)\ =\ 1\">\r<BR>\n" );
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document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_4(4)\"> is the probability that NONE are late, so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 1\ -\ P_4(4)\">\r<BR>\n" );
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document.write( "is the probability that at least 1 is late.  This is a simple subtraction since you already have the value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_4(4)\"> from question 1.\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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