document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1152675>Question 1152675</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The rates of on-time flights for commercial jets are continuously tracked by the U.S. Department of Transportation. Recently, Southwest Air had the best rate with 80 % of its flights arriving on time. A test is conducted by randomly selecting 14 Southwest flights and observing whether they arrive on time. Find the probability that at least 12 flights arrive on time. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #774741 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=774741\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "p = 0.8 = probabiliity of success = probability of flight on time<BR>\n" );
document.write( "n = 14 = sample size\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Either the flight is on time (success) or it is not on time (failure). Let's assume, for the sake of simplicity, that each flight is independent of any other flight. Based on that, this means we have a binomial problem. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We will use the binomial PDF function<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28x%29+=+%28%28n%21%29%2F%28x%21%28n-x%29%21%29%29%2A%28p%5Ex%29%2A%281-p%29%5E%28n-x%29 ALIGN=MIDDLE  ALT=\"B%28x%29+=+%28%28n%21%29%2F%28x%21%28n-x%29%21%29%29%2A%28p%5Ex%29%2A%281-p%29%5E%28n-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to compute individual binomial probabilities. <BR>\n" );
document.write( "The expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%21%29%2F%28x%21%28n-x%29%21%29 ALIGN=MIDDLE  ALT=\"%28n%21%29%2F%28x%21%28n-x%29%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents the binomial coefficient. It can be found alternatively through Pascal's Triangle. The exclamation marks represent factorials.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Plug in n = 14 and p = 0.8 to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28x%29+=+%28%28n%21%29%2F%28x%21%28n-x%29%21%29%29%2A%28p%5Ex%29%2A%281-p%29%5E%28n-x%29 ALIGN=MIDDLE  ALT=\"B%28x%29+=+%28%28n%21%29%2F%28x%21%28n-x%29%21%29%29%2A%28p%5Ex%29%2A%281-p%29%5E%28n-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28x%29+=+%28%2814%21%29%2F%28x%21%2814-x%29%21%29%29%2A%280.8%5Ex%29%2A%281-0.8%29%5E%2814-x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "From here we need to compute B(x) for x = 12, 13 and 14. <BR>\n" );
document.write( "Then we'll add up those results.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Plug in x = 12\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28x%29+=+%28%2814%21%29%2F%28x%21%2814-x%29%21%29%29%2A%280.8%5Ex%29%2A%281-0.8%29%5E%2814-x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%2812%29+=+%28%2814%21%29%2F%2812%21%2814-12%29%21%29%29%2A%280.8%5E12%29%2A%281-0.8%29%5E%2814-12%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%2812%29+=+0.25014 ALIGN=MIDDLE  ALT=\"B%2812%29+=+0.25014\"   DISABLED_style= \"max-width:90%; height:auto;\" > This represents the approximate probability of exactly 12 flights are on time. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Plug in x = 13\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28x%29+=+%28%2814%21%29%2F%28x%21%2814-x%29%21%29%29%2A%280.8%5Ex%29%2A%281-0.8%29%5E%2814-x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28x%29+=+%28%2814%21%29%2F%2813%21%2814-13%29%21%29%29%2A%280.8%5E13%29%2A%281-0.8%29%5E%2814-13%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%2813%29+=+0.15393 ALIGN=MIDDLE  ALT=\"B%2813%29+=+0.15393\"   DISABLED_style= \"max-width:90%; height:auto;\" > This represents the approximate probability of exactly 13 flights are on time. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Plug in x = 14\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28x%29+=+%28%2814%21%29%2F%28x%21%2814-x%29%21%29%29%2A%280.8%5Ex%29%2A%281-0.8%29%5E%2814-x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28x%29+=+%28%2814%21%29%2F%2814%21%2814-14%29%21%29%29%2A%280.8%5E14%29%2A%281-0.8%29%5E%2814-14%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%2814%29+=+0.04398 ALIGN=MIDDLE  ALT=\"B%2814%29+=+0.04398\"   DISABLED_style= \"max-width:90%; height:auto;\" > This represents the approximate probability of exactly 14 flights are on time. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Add up the results to find the probability that x is 12 or larger.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28x+%3E=+12%29+=+B%2812%29+%2B+B%2813%29+%2B+B%2814%29 ALIGN=MIDDLE  ALT=\"B%28x+%3E=+12%29+=+B%2812%29+%2B+B%2813%29+%2B+B%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28x+%3E=+12%29+=+0.25014+%2B+0.15393+%2B+0.04398 ALIGN=MIDDLE  ALT=\"B%28x+%3E=+12%29+=+0.25014+%2B+0.15393+%2B+0.04398\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28x+%3E=+12%29+=+0.44805 ALIGN=MIDDLE  ALT=\"B%28x+%3E=+12%29+=+0.44805\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------------------------------------------------------<BR>\n" );
document.write( "----------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<font size=4 color=red>Answer: approximately 0.44805</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );