document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205465>Question 1205465</A>: <I><SPAN STYLE=\"word-break: break-all;\"> User<BR>\n" );
document.write( "A small regional carrier accepted 16 reservations for a particular flight with 14 seats. 10 reservations went to regular customers who will arrive for the flight. Each of the remaining passengers will arrive for the flight with a 47% chance, independently of each other.<BR>\n" );
document.write( "(Report answers accurate to 4 decimal places.)\r<BR>\n" );
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document.write( "Find the probability that overbooking occurs. <BR>\n" );
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document.write( "Find the probability that the flight has empty seats\r<BR>\n" );
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document.write( "A similar question has been asked before. <BR>\n" );
document.write( "Check out this link<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1204038.html\">https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1204038.html</a>\r<BR>\n" );
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document.write( "If you're still stuck, then read on.\r<BR>\n" );
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document.write( "There are 14 seats. Ten of which are locked in for those guaranteed to show up for the flight.<BR>\n" );
document.write( "There are 14-10 = 4 seats left and 16-10 = 6 extra people competing for those remaining seats.<BR>\n" );
document.write( "These 6 extra people may or may not show up on time. \r<BR>\n" );
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document.write( "n = 6 = sample size<BR>\n" );
document.write( "p = 0.47 = probability a person arrives on time<BR>\n" );
document.write( "x = number extra people who arrive on time<BR>\n" );
document.write( "x is equal to a value from the set {0,1,2,3,4,5,6}<BR>\n" );
document.write( "x = 2 for instance means 2 extra people arrived on time.\r<BR>\n" );
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document.write( "Random variable X follows a binonimal distribution because of these 3 reasons:<ul><li>There are two outcomes: the person shows up on time or they don't. The 2 outcomes is what the \"bi\" in \"binomial\" refers to.</li><li>Each person has the same probability of showing up on time (47% chance).</li><li>Each person's arrival is independent of one another. This assumes no one person hinders or helps any other potential passenger.</li></ul>Always check these 3 criteria to make sure we are dealing with a binomial probability distribution.\r<BR>\n" );
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document.write( "B(x) = binomial probability<BR>\n" );
document.write( "B(x) = (nCx)*(p^x)*(1-p)^(n-x)<BR>\n" );
document.write( "The nCx refers to the nCr combination formula. Such values can be found in Pascal's Triangle.\r<BR>\n" );
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document.write( "Plug in the values mentioned for n and p<BR>\n" );
document.write( "We'll go from this<BR>\n" );
document.write( "B(x) = (nCx)*(p^x)*(1-p)^(n-x)<BR>\n" );
document.write( "to this<BR>\n" );
document.write( "B(x) = (6Cx)*(0.47^x)*(0.53)^(6-x)\r<BR>\n" );
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document.write( "Then let's plug in x = 0<BR>\n" );
document.write( "B(x) = (6Cx)*(0.47^x)*(0.53)^(6-x)<BR>\n" );
document.write( "B(0) = (6C0)*(0.47^0)*(0.53)^(6-0)<BR>\n" );
document.write( "B(0) = (1)*(0.47^0)*(0.53)^(6-0)<BR>\n" );
document.write( "B(0) = 0.022164361129<BR>\n" );
document.write( "B(0) = 0.0222<BR>\n" );
document.write( "This is the approximate probability of having 0 extra people show up.\r<BR>\n" );
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document.write( "You could repeat similar steps for x = 1 through x = 6.\r<BR>\n" );
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document.write( "A quicker way is to use a spreadsheet. Specifically you'll use the <a  rel=nofollow HREF=\"https://support.microsoft.com/en-us/office/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c\">binomDist</a> function. <BR>\n" );
document.write( "Then use the <a  rel=nofollow HREF=\"https://support.microsoft.com/en-us/office/round-function-c018c5d8-40fb-4053-90b1-b3e7f61a213c\">round</a> function to do as you'd expect.<BR>\n" );
document.write( "Example:  <BR>\n" );
document.write( "<font color=green>=Round(A1,4)</font> will round the value at cell A1 to 4 decimal places. Don't forget about the equal sign up front. \r<BR>\n" );
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document.write( "Here's the binomial probability distribution<BR>\n" );
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document.write( "<tr><TD >x</TD><TD >B(x)</TD></tr><tr><TD >0</TD><TD >0.0222</TD></tr><tr><TD >1</TD><TD >0.1179</TD></tr><tr><TD >2</TD><TD >0.2615</TD></tr><tr><TD >3</TD><TD >0.3091</TD></tr><tr><TD >4</TD><TD >0.2056</TD></tr><tr><TD >5</TD><TD >0.0729</TD></tr><tr><TD >6</TD><TD >0.0108</TD></tr>\n" );
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document.write( "Overbooking happens when x = 5 or x = 6 extra people show up on time.<BR>\n" );
document.write( "B(5) + B(6) = 0.0729+0.0108 <BR>\n" );
document.write( "B(5) + B(6) = <font color=red>0.0837</font> is the approximate probability of overbooking. <BR>\n" );
document.write( "This is roughly a 8.37% chance.\r<BR>\n" );
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document.write( "Empty seats happen when x = 0 through x = 3, i.e when x < 4.<BR>\n" );
document.write( "B(0)+B(1)+B(2)+B(3) = 0.0222+0.1179+0.2615+0.3091 <BR>\n" );
document.write( "B(0)+B(1)+B(2)+B(3) = <font color=red>0.7107</font><BR>\n" );
document.write( "There's roughly a 71.07% chance of the flight having one or more empty seats.\r<BR>\n" );
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document.write( "Answers:\r<BR>\n" );
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document.write( "Probability that overbooking occurs:  <font color=red size=4>0.0837</font><BR>\n" );
document.write( "Probability that the flight has empty seats: <font color=red size=4>0.7107</font><BR>\n" );
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