document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1083199>Question 1083199</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Because not all air line passengers show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 120 passengers.the probability that a passenger doesn't show up is 0.10, and the passengers behave independently.<BR>\n" );
document.write( "   a/ what is the probability that every passenger who show up can take the flight?<BR>\n" );
document.write( "   b/ what is the probability that the flight departs with empty seats?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #697230 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=697230\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 125 bought tickets.<BR>\n" );
document.write( "only 120 seats available.\r<BR>\n" );
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document.write( "the probability that a person who bought a ticket will not show up is .1.\r<BR>\n" );
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document.write( "therefore, the probability that a person who bought a ticket does show up is 1 minus .1 which is equal to .9.\r<BR>\n" );
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document.write( "if less than 121 people show up, then everybody gets a seat.<BR>\n" );
document.write( "that's because 120 or less show up and there are 120 seats available.\r<BR>\n" );
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document.write( "if less than 120 people show up, then there will be at least one empty seat.<BR>\n" );
document.write( "that because 119 or less show up and there are 120 seats available.\r<BR>\n" );
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document.write( "when doing the calculations manually, you want to exert as little effort as possible.\r<BR>\n" );
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document.write( "the probability that less than 121 people show up is equal to 1 minus the probability that 121 or more show up.\r<BR>\n" );
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document.write( "the probability that 121 or more people show up is equal to the sum of p(121) + p(122) + p(123) + p(124) + p(125).\r<BR>\n" );
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document.write( "the binomial probability formula is p(x) = c(n,x) * p^x * q^(n-x)\r<BR>\n" );
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document.write( "x is the number of successes and can range from 0 to 125 in this problem.<BR>\n" );
document.write( "p is the probability of success which is .9 since the probability that a person will show up is .9.<BR>\n" );
document.write( "q is the probability of failure which is .1 since the probability that a person will not show up is .1.<BR>\n" );
document.write( "c(n,x) is the number of ways you can get sets of x elements each out of a set of n elements.<BR>\n" );
document.write( "that formula for c(n,x)is c(n,x) = n! / (x! * (n-x)!)\r<BR>\n" );
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document.write( "i used excel to do the calculations.\r<BR>\n" );
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document.write( "an example of one of the calculations would be p(121) = c(125,121) * .9^121 * .1^4.\r<BR>\n" );
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document.write( "c(125,121) is equal to 9691375<BR>\n" );
document.write( ".9^121 is equal to 2.906321 * 10^-6<BR>\n" );
document.write( ".1^4 is equal to 1 * 10^-4\r<BR>\n" );
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document.write( "p(121) is therefore equal to 9691375 * 2.906321 * 10^-6 * 1 * 10^-4.<BR>\n" );
document.write( "the result is p(121) = .002816625.\r<BR>\n" );
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document.write( "a picture of the excel printout, shown below, confirms this.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/053001.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the probability that everybody who shows up will get a seat is the same as the probability that less than 121 people will show up.\r<BR>\n" );
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document.write( "the probability that less than 121 people show up is equal to 1 minus the the probability that 121 or more people show up.\r<BR>\n" );
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document.write( "the probability that 121 or more people show up is the sum of p(121) through p(125) as shown in the following excel printout.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/053002.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the sum of p(121) through p(125) is equal to 0.0039 rounded to 4 decimal places as shown in that same printout.\r<BR>\n" );
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document.write( "1 minus the sum of p(121) through p(125) is equal to 0.9961 rounded to 4 decimal places as also shown in that same printout.\r<BR>\n" );
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document.write( "the probability that there will be at least one empty seat is the probability that less than 120 people will show up.\r<BR>\n" );
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document.write( "this is equal to 1 minus the probability that 120 or more people will show up.\r<BR>\n" );
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document.write( "the probability that 120 or more people show up is the sum of p(120) through p(125) as shown in the following excel printout.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/053003.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the sum of p(120) through p(125) is equal to 0.0114 rounded to 4 decimal places as shown in that same printout.\r<BR>\n" );
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document.write( "1 minus the sum of p(120) though p(125) is equal to 0.9886 rounded to 4 decimal places as also shown in that same printout.\r<BR>\n" );
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document.write( "you were asked the following questions.\r<BR>\n" );
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document.write( "a/ what is the probability that every passenger who show up can take the flight?\r<BR>\n" );
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document.write( "the answer to this is that the probability that every passenger who shows up can take the flight is .9961 or 99.61%.\r<BR>\n" );
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document.write( "b/ what is the probability that the flight departs with empty seats?\r<BR>\n" );
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document.write( "the answer to this is that the probability that the flight departs with empty seats is .9886 or 98.86%.\r<BR>\n" );
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document.write( "this all assumes that i did this correctly, which i believe i did, but there's always the probability that i thought i did it correctly but actually didn't.\r<BR>\n" );
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document.write( "i'm guessing i did it correctly this time.\r<BR>\n" );
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document.write( "you can also use an online binomial calculator to provide these results, and to confirm that you did it correctly.\r<BR>\n" );
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document.write( "one such calculator can be found here.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://onlinestatbook.com/2/calculators/binomial_dist.html\" target = \"_blank\">http://onlinestatbook.com/2/calculators/binomial_dist.html</a>\r<BR>\n" );
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document.write( "the main this is to make sure that your assumptions are correct.\r<BR>\n" );
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document.write( "using this calculator, i did the following:\r<BR>\n" );
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document.write( "n is equal to 125 because that's the number of tickets that were sold.\r<BR>\n" );
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document.write( "p is .9 because that's the probability that a person who bought a ticket will show up for the flight.\r<BR>\n" );
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document.write( "if the number of people who show up is less than 121, then all of the people who show up will get a seat because 120 or less will show up and there are 120 seats on the plane.\r<BR>\n" );
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document.write( "put 121 in the less than field and click recalculate and the calculator will tell you that the probability is .9961.\r<BR>\n" );
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document.write( "if the number of people who show up is less than 120, then there will be at least 1 empty seat.\r<BR>\n" );
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document.write( "that's because 119 or less show up and there are 120 seats available.\r<BR>\n" );
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document.write( "put 120 in the less than field and click recalculate and the calculator will tell you that the probability is .9886.\r<BR>\n" );
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document.write( "this agrees with what i calculated in excel and took a lot less work in setting up that i had to when i did the calculations in excel.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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