document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177125>Question 1177125</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 7. In Alberta, 55% of new drivers take a driver training course before completing their driver’s test. Of the drivers that took training, 75% pass their driver’s test. Of the drivers that did not take training, 58% pass their driver’s test.\r<BR>\n" );
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document.write( "a. Complete the following tree diagram to represent the given information. Express the probability of each event to 4 decimal places if necessary. \r<BR>\n" );
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document.write( "Diagram: -------> i= P(training)------>iii= P(pass)----> vii= P(training AND pass)<BR>\n" );
document.write( "                                -------> iv= P(fail)----> viii= P(training AND fail)    \r<BR>\n" );
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document.write( "        <BR>\n" );
document.write( "         -------> ii= P(no training)-----> v= P(pass)-----> ix=P(no training AND pass)<BR>\n" );
document.write( "                                  ------> vi= P(fail)----> x= P(no training AND fail) \r<BR>\n" );
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document.write( "MY ANSWERS: <BR>\n" );
document.write( "i.	 55%	<BR>\n" );
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document.write( "ii.	 45%	<BR>\n" );
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document.write( "iii.	 75%	<BR>\n" );
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document.write( "iv.	 25%	<BR>\n" );
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document.write( "v.	 58%	<BR>\n" );
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document.write( "vi.      42	<BR>\n" );
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document.write( "vii.     0.42	<BR>\n" );
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document.write( "viii.    0.14	<BR>\n" );
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document.write( "ix.      0.26	<BR>\n" );
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document.write( "x.	 0.18	<BR>\n" );
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document.write( "b.	Jim passed his driver’s test. Use the values in your probability tree above to determine the probability that he took driver training, to the nearest whole percent? (1 mark)<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 #850517 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850517\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It looks like you've got a good handle on the tree diagram and probabilities! Here's a breakdown with some minor corrections:\r<BR>\n" );
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document.write( "**a. Completed Tree Diagram**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "                                  ---  P(Pass | Training) = 0.75  --> P(Training AND Pass) = 0.55 * 0.75 = 0.4125 <BR>\n" );
document.write( "                      /           <BR>\n" );
document.write( "                     /<BR>\n" );
document.write( "       P(Training) = 0.55          <BR>\n" );
document.write( "                    \              --- P(Fail | Training) = 0.25 --> P(Training AND Fail) = 0.55 * 0.25 = 0.1375<BR>\n" );
document.write( "                     \<BR>\n" );
document.write( "                      /<BR>\n" );
document.write( "                                  --- P(Pass | No Training) = 0.58 --> P(No Training AND Pass) = 0.45 * 0.58 = 0.2610<BR>\n" );
document.write( "                      \           <BR>\n" );
document.write( "                     /<BR>\n" );
document.write( "       P(No Training) = 0.45        <BR>\n" );
document.write( "                    \              --- P(Fail | No Training) = 0.42 --> P(No Training AND Fail) = 0.45 * 0.42 = 0.1890<BR>\n" );
document.write( "                     \<BR>\n" );
document.write( "                      / <BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Minor Corrections:**\r<BR>\n" );
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document.write( "* **vii. P(Training AND Pass):** 0.55 * 0.75 = 0.4125 <BR>\n" );
document.write( "* **viii. P(Training AND Fail):** 0.55 * 0.25 = 0.1375<BR>\n" );
document.write( "* **ix. P(No Training AND Pass):** 0.45 * 0.58 = 0.2610<BR>\n" );
document.write( "* **x. P(No Training AND Fail):** 0.45 * 0.42 = 0.1890\r<BR>\n" );
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document.write( "You had the right idea with the calculations, just a slight adjustment in the values for vii and viii.\r<BR>\n" );
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document.write( "**b. Probability Jim Took Training Given He Passed**\r<BR>\n" );
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document.write( "We want to find P(Training | Pass). We can use Bayes' Theorem:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "P(Training | Pass) = [P(Pass | Training) * P(Training)] / P(Pass) <BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "To find P(Pass), we need to consider both ways someone can pass:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "P(Pass) = P(Training AND Pass) + P(No Training AND Pass)<BR>\n" );
document.write( "P(Pass) = 0.4125 + 0.2610 = 0.6735<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Now we can plug the values into Bayes' Theorem:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "P(Training | Pass) = (0.75 * 0.55) / 0.6735 <BR>\n" );
document.write( "                     ≈ 0.6126<BR>\n" );
document.write( "                     ≈ 61% (to the nearest whole percent)<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Therefore, the probability that Jim took driver training given that he passed his test is approximately 61%.** </SPAN>\n" );
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