document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206968>Question 1206968</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A researcher wishes to conduct a study of the color preference of new car buyers. Suppose that 30% of this population prefers the color brown. If 14 buyers are randomly selected, what is the probability that at least 3 buyers would prefer brown? Round your answer to the nearest ten-thousandth, if necessary.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #844721 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=844721\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer:  <font color=red size=4>0.8392</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "Binomial formula<BR>\n" );
document.write( "B(x) = (nCx)*(p^x)*(1-p)^(n-x)<BR>\n" );
document.write( "where,<BR>\n" );
document.write( "n = sample size = 14<BR>\n" );
document.write( "p = probability a person prefers a brown car = 0.30<BR>\n" );
document.write( "x = number of people who prefer a brown car<BR>\n" );
document.write( "x spans from x = 0 to x = 14.<BR>\n" );
document.write( "i.e. x is from the set {0,1,2,..,13,14}\r<BR>\n" );
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document.write( "The nCx refers to the nCr combination formula. <BR>\n" );
document.write( "Some textbooks call it the \"choose\" formula. \r<BR>\n" );
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document.write( "Let's compute the binomial probability that x = 0 people prefer a brown car.<BR>\n" );
document.write( "B(x) = (nCx)*(p^x)*(1-p)^(n-x)<BR>\n" );
document.write( "B(x) = (14Cx)*(0.3^x)*(1-0.3)^(14-x)<BR>\n" );
document.write( "B(0) = (14C0)*(0.3^0)*(0.7)^(14-0)<BR>\n" );
document.write( "B(0) = (1)*(0.3^0)*(0.7)^(14-0)<BR>\n" );
document.write( "B(0) = 0.00678223 which is approximate<BR>\n" );
document.write( "There's about a 0.678% chance of 0 people in the sample preferring a brown car (i.e. that everyone in the sample prefers some other color than brown).\r<BR>\n" );
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document.write( "Follow similar steps for x = 1 and x = 2 to find:<BR>\n" );
document.write( "B(1) = 0.04069338<BR>\n" );
document.write( "B(2) = 0.11336014<BR>\n" );
document.write( "These values are approximate.\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "B(0)+B(1)+B(2) = 0.00678223 + 0.04069338 + 0.11336014 = 0.16083575<BR>\n" );
document.write( "There's about a 16.08% chance of selecting between x = 0 and x = 2 people who prefer a brown car.\r<BR>\n" );
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document.write( "Subtract this from 1 to get the final answer. This is because:<BR>\n" );
document.write( "<font color=blue>B(0)+B(1)+B(2)</font> + <font color=red>B(3)+B(4)+B(5)+...+B(13)+B(14)</font> = 1<BR>\n" );
document.write( "which leads to<BR>\n" );
document.write( "<font color=red>B(3)+B(4)+B(5)+...+B(13)+B(14)</font>  = 1- ( <font color=blue>B(0)+B(1)+B(2)</font> )<BR>\n" );
document.write( "The stuff in red is the answer we want while the stuff in blue is what we calculated earlier.\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "1 - 0.16083575 = 0.83916425<BR>\n" );
document.write( "This rounds to <font color=red>0.8392</font> when rounding to the nearest ten-thousandth aka rounding to 4 decimal places.\r<BR>\n" );
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document.write( "--------------------------------------------------------------------------------------------------------------------------\r<BR>\n" );
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document.write( "As the other tutor mentions, there are fast shortcuts using technology.\r<BR>\n" );
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document.write( "TI83 or TI84:<BR>\n" );
document.write( "<font color=red>1 - binomCDF(14, 0.3, 2)</font><BR>\n" );
document.write( "The binomCDF function is found by pressing the button labeled \"2nd\" and then the VARS key. <BR>\n" );
document.write( "Scroll down a bit until you find the function. \r<BR>\n" );
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document.write( "Spreadsheet:<BR>\n" );
document.write( "<font color=red>=1-BinomDist(2,14,0.3,1)</font><BR>\n" );
document.write( "Don't forget about the equal sign up front. If that isn't typed in, then the spreadsheet will think you are entering plain text, and the formula will not evaluate. <BR>\n" );
document.write( "The \"1\" at the end of the command tells the spreadsheet to use a CDF rather than a PDF.<BR>\n" );
document.write( "Refer to the help manual for more information.\r<BR>\n" );
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document.write( "GeoGebra:<BR>\n" );
document.write( "Open up the probability calculator viewport and select \"binomial\" under the list of distributions. <BR>\n" );
document.write( "Type in the parameters n = 14, p = 0.3 and you should see a table of values show up. <BR>\n" );
document.write( "Select values x = 3 through x = 14 to have it add up the proper B(x) values we want.<BR>\n" );
document.write( "Or you can type 3 and 14 into the input boxes that correspond to P(__ < x < ___) \r<BR>\n" );
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document.write( "There are many online calculators that are great alternatives.<BR>\n" );
document.write( "Feel free to explore your favorite.<BR>\n" );
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