document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=880388>Question 880388</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Me again.\r<BR>\n" );
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document.write( "Hi - I know this is probably really simple, are I do not know if I am on the right track for this question (probability is just not making any sense to me at all).  \r<BR>\n" );
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document.write( "It is known that 25% of all staff in a company make use of the free influenza vaccination programme. Answer the following as:  The probability is equal to:  (Round your probability to 4 decimal places:\r<BR>\n" );
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document.write( "Q  If 20 people are randomly selected, find the probability that exactly 12 people don't use the vaccination programme.\r<BR>\n" );
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document.write( "Please help!<BR>\n" );
document.write( "Thanks in advance! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #531446 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=531446\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I think we are all reading it wrong.\r<BR>\n" );
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document.write( "THE ANSWER:<BR>\n" );
document.write( "Maybe you just want the answer. In that case, I calculated the approximate probability that 12 of those 20 people got vaccinated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2220+C12%220.25%5E12%2A0.75%5E8=0.0008 ALIGN=MIDDLE  ALT=\"%2220+C12%220.25%5E12%2A0.75%5E8=0.0008\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but I think you may really have meant what is the probability that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" > were not vaccinated.<BR>\n" );
document.write( "Then, it would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2220+C12%220.25%5E8%2A0.75%5E12=0.0609 ALIGN=MIDDLE  ALT=\"%2220+C12%220.25%5E8%2A0.75%5E12=0.0609\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "THE CALCULATION:<BR>\n" );
document.write( "Maybe you want to know how to calculate it.<BR>\n" );
document.write( "If you have a computer with the right software, or a calculator with enough functions, you can calculate combinations of 20 taking 12 at a time as 20C12=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C20%2C12%29%29=125970 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2C20%2C12%29%29=125970\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With a simpler calculator, or (gasp!) with pencil and paper,<BR>\n" );
document.write( "you would calculate it as<BR>\n" );
document.write( "20C12=20C8=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20%2A19%2A18%2A17%2A16%2A15%2A14%2F%281%2A2%2A3%2A4%2A5%2A6%2A7%2A8%29=2%2A10%2A19%2A3%2A6%2A17%2A4%2A4%2A3%2A5%2A2%2A7%2A13%2F%281%2A2%2A3%2A4%2A5%2A6%2A7%2A2%2A4%29=10%2A19%2A3%2A17%2A13%2F1=10%2A19%2A17%2A13%2A3=10%2A323%2A39=12597 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With most calculators or using your computer you can calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.25%5E8=1.5259%2A10%5E%28-5%29 ALIGN=MIDDLE  ALT=\"0.25%5E8=1.5259%2A10%5E%28-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (approximately) and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.75%5E12=0.031676 ALIGN=MIDDLE  ALT=\"0.75%5E12=0.031676\"   DISABLED_style= \"max-width:90%; height:auto;\" > (approximately).<BR>\n" );
document.write( "Then you calculate your approximate probability as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125970%2A5.96%2A10%5E%28-8%29%2A0.100=0.06088685426148 ALIGN=MIDDLE  ALT=\"125970%2A5.96%2A10%5E%28-8%29%2A0.100=0.06088685426148\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which rounds to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.0609 ALIGN=MIDDLE  ALT=\"0.0609\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With pencil and paper it gets a little more cumbersome.<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE REASON WHY YOU APPLY \"THE BINOMIAL DISTRIBUTION\":<BR>\n" );
document.write( "Maybe you want to understand why.<BR>\n" );
document.write( "You can describe all the possible outcomes of randomly picking <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the people in the staff of that company by the binomial<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.25v%2B0.75n ALIGN=MIDDLE  ALT=\"0.25v%2B0.75n\"   DISABLED_style= \"max-width:90%; height:auto;\" > , meaning that<BR>\n" );
document.write( "there is  a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.75 ALIGN=MIDDLE  ALT=\"0.75\"   DISABLED_style= \"max-width:90%; height:auto;\" > probability that he/she got the vaccine and<BR>\n" );
document.write( "there is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.75 ALIGN=MIDDLE  ALT=\"0.75\"   DISABLED_style= \"max-width:90%; height:auto;\" > probability that he/she did not get the vaccine.<BR>\n" );
document.write( "If you pick <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > people, you can describe all the possible outcomes as the square of that binomial:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280.25v%2B0.75n%29%280.25v%2B0.75n%29=%280.25v%2B0.75n%29%5E2=%280.25v%29%5E2%2B2%2A%280.25%29%280.75%29vn%2B%280.75%29n%5E2=0.25%5E2v%5E2%2B0.375nv%2B%280.75%29n%5E2=0.0625v%5E2%2B0.375nv%2B0.5625n%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That means there is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.0625 ALIGN=MIDDLE  ALT=\"0.0625\"   DISABLED_style= \"max-width:90%; height:auto;\" > probability that both goth the vaccine;<BR>\n" );
document.write( "there is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.375 ALIGN=MIDDLE  ALT=\"0.375\"   DISABLED_style= \"max-width:90%; height:auto;\" > probability that one or the other (got the vaccine(but not both),<BR>\n" );
document.write( "and there is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.5625 ALIGN=MIDDLE  ALT=\"0.5625\"   DISABLED_style= \"max-width:90%; height:auto;\" > probability that both are unvaccinated.<BR>\n" );
document.write( "If you pick <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" > people, the whole spectrum of probabilities is described by the 20th power of that binomial:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280.25v%2B0.75n%29%5E20=0.25%5E20v%5E20%2B20%2A0.25%5E19%2A0.75v%5E19%2An%2B190%2A0.25%5E18%2A0.75%5E2v%5E18n%5E2%2B+%22...%22+%2B%28matrix%282%2C1%2C10%2C12%29%290.25%5E8%2A0.75%5E12%2Av%5E8n%5E12%2B+%22...%22+%2B190%2A0.25%5E2%2A0.75%5E18%2Av%5E2%2An%5E18%2B20%2A0.25%2A0.75%5E19%2Av%2An%5E18%2B0.75%5E20n%5E20 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In that long polynomial, the coefficient of the term with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5E12 ALIGN=MIDDLE  ALT=\"v%5E12\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents the probability that exactly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > of those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" > people were vaccinated, and the other <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > did not use he vaccination programme. </SPAN>\n" );
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