document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192508>Question 1192508</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (1 point) A breathalyser test is used by police in an area to determine whether a driver has an excess of alcohol in their blood. The device is not totally reliable: 7 % of drivers who have not consumed an excess of alcohol give a reading from the breathalyser as being above the legal limit, while 10 % of drivers who are above the legal limit will give a reading below that level. Suppose that in fact 14 % of drivers are above the legal alcohol limit, and the police stop a driver at random. Give answers to the following to four decimal places.\r<BR>\n" );
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document.write( "Part a)<BR>\n" );
document.write( "What is the probability that the driver is incorrectly classified as being over the limit?\r<BR>\n" );
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document.write( "Part b)<BR>\n" );
document.write( "What is the probability that the driver is correctly classified as being over the limit?\r<BR>\n" );
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document.write( "Part c)<BR>\n" );
document.write( "Find the probability that the driver gives a breathalyser test reading that is over the limit.\r<BR>\n" );
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document.write( "Part d)<BR>\n" );
document.write( "Find the probability that the driver is under the legal limit, given the breathalyser reading is also below the limit.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849029 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52946)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849029\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "A breathalyzer test is used by police in an area to determine whether a driver has an excess of alcohol in their blood. <BR>\n" );
document.write( "The device is not totally reliable: 7% of drivers who have not consumed an excess of alcohol give a reading <BR>\n" );
document.write( "from the breathalyzer as being above the legal limit, while 10% of drivers who are above the legal limit <BR>\n" );
document.write( "will give a reading below that level. Suppose that in fact 14% of drivers are above the legal alcohol limit, <BR>\n" );
document.write( "and the police stop a driver at random. Give answers to the following to four decimal places.\r<BR>\n" );
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document.write( "(a) What is the probability that the driver is incorrectly classified as being over the limit?\r<BR>\n" );
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document.write( "(b) What is the probability that the driver is correctly classified as being over the limit?\r<BR>\n" );
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document.write( "(c) Find the probability that the driver gives a breathalyzer test reading that is over the limit.\r<BR>\n" );
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document.write( "(d) Find the probability that the driver is under the legal limit, given the breathalyzer reading is also below the limit.<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In the post by @CPhill, &nbsp;parts &nbsp;(a), &nbsp;(b) &nbsp;and &nbsp;(d) &nbsp;are solved and answered &nbsp;INCORRECTLY.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I came to bring correct solutions to all parts.\r<BR>\n" );
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document.write( "(a)  In (a), they want you determine the probability of two simultaneous events:\r\n" );
document.write( "     the driver is under the limit, but the reading incorrectly classified him\r\n" );
document.write( "     as being over the limit.\r\n" );
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document.write( "     The <U>ANSWER</U> to (a) is  (1-0.14)*0.07 = 0.0602 = 6.02%.\r\n" );
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document.write( "(b)  In (b), they want you determine the probability of two simultaneous events:\r\n" );
document.write( "     the driver is over the limit, and the reading correctly classified him \r\n" );
document.write( "     as being over the limit.\r\n" );
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document.write( "     The <U>ANSWER</U> to (b) is  0.14*(1-0.1) = 0.126 = 12.6%.   \r\n" );
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document.write( "(c)  This probability is the sum of two probabilities of disjoint events:\r\n" );
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document.write( "         - the driver is under the limit (1-0.14), but the reading is mistakenly over the limit\r\n" );
document.write( "               P1 = (1-0.14)*0.07;\r\n" );
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document.write( "         - the driver is over  the limit (0.14),   and the reading is correctly over the limit\r\n" );
document.write( "               P2 = 0.14*0.9.\r\n" );
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document.write( "      Therefore, the probability in (c) is  P = P1 + P2= (1-0.14)*0.07 + 0.14*0.9 = 0.1862 = 18.62%.  <U>ANSWER</U> to (c)\r\n" );
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document.write( "(d)  In (d), the conditional probability is \r\n" );
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document.write( "          P = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281-0.14%29%2A%281-0.07%29%29%2F%28%281-0.14%29%2A%281-0.07%29%2B0.14%2A%281-0.9%29%29 ALIGN=MIDDLE  ALT=\"%28%281-0.14%29%2A%281-0.07%29%29%2F%28%281-0.14%29%2A%281-0.07%29%2B0.14%2A%281-0.9%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.7998%2F0.8138 ALIGN=MIDDLE  ALT=\"0.7998%2F0.8138\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 0.9830 = 98.30%.  <U>ANSWER</U> to (d)\r\n" );
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document.write( "     In this formula, the numerator is the probability of the event that the driver is under the legal limit \r\n" );
document.write( "     and the reading is under the legal limit.\r\n" );
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document.write( "     The denominator is the probability that the reading is below the legal limit.\r\n" );
document.write( "     The structure of the denominator here is similar to expression in part (c).\r\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "The goal of this problem is to teach a student to think logically.<BR>\n" );
document.write( "Having and using common sense is enough to make every step.\r<BR>\n" );
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document.write( "The &nbsp;TWIN &nbsp;to this problem was solved at this forum many years ago under this link\r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1206991.html\r<BR>\n" );
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document.write( "This example on how incorrectly the problem was solved in the post by @CPhill shows<BR>\n" );
document.write( "that the references to formal theorems do not save from making monstrous errors.\r<BR>\n" );
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document.write( "Common sense should work and really works much better in such simple problems.<BR>\n" );
document.write( "And this is exactly my major goal in this post to develop the student's common sense.\r<BR>\n" );
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document.write( "The last answer in the post by @CPhill\r<BR>\n" );
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document.write( "* d) Probability of driver under the limit given breathalyser reading below limit: 0.0740\r<BR>\n" );
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document.write( "tells me that, in reality, nobody even read it and nobody even thought on/about it <BR>\n" );
document.write( "before submitting it to the forum.\r<BR>\n" );
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document.write( "It is below any common sense level.\r<BR>\n" );
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