document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177086>Question 1177086</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A consumer goods company recruits several graduating students from universities each year. Concerned about the high cost of training new employees, the company instituted a review of attrition among new recruits. Over five years, 30% of new recruits came from a local university, and the balance came from a more distant universities. Of the new recruits, 20% of those who were students from a local university resigned within two years, while 45% of other students resigned. Given that a student resigned within two years, what is the probability that she hired from<BR>\n" );
document.write( "a) a local university?<BR>\n" );
document.write( "b) a more distant university? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850526 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=850526\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely, let's break down this problem using Bayes' Theorem.\r<BR>\n" );
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document.write( "**Understanding the Problem**\r<BR>\n" );
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document.write( "* **Events:**<BR>\n" );
document.write( "    * L: Recruited from a local university.<BR>\n" );
document.write( "    * D: Recruited from a distant university.<BR>\n" );
document.write( "    * R: Resigned within two years.<BR>\n" );
document.write( "* **Given Probabilities:**<BR>\n" );
document.write( "    * P(L) = 0.30 (30% from local)<BR>\n" );
document.write( "    * P(D) = 0.70 (70% from distant)<BR>\n" );
document.write( "    * P(R|L) = 0.20 (20% of local resigned)<BR>\n" );
document.write( "    * P(R|D) = 0.45 (45% of distant resigned)\r<BR>\n" );
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document.write( "**What We Need to Find**\r<BR>\n" );
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document.write( "* a) P(L|R): Probability of being from a local university given resignation.<BR>\n" );
document.write( "* b) P(D|R): Probability of being from a distant university given resignation.\r<BR>\n" );
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document.write( "**Applying Bayes' Theorem**\r<BR>\n" );
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document.write( "Bayes' Theorem states: P(A|B) = [P(B|A) * P(A)] / P(B)\r<BR>\n" );
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document.write( "In our case:\r<BR>\n" );
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document.write( "* P(L|R) = [P(R|L) * P(L)] / P(R)<BR>\n" );
document.write( "* P(D|R) = [P(R|D) * P(D)] / P(R)\r<BR>\n" );
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document.write( "**Calculating P(R) (Total Probability of Resignation)**\r<BR>\n" );
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document.write( "We need to find the overall probability of resignation, P(R). We can do this using the law of total probability:\r<BR>\n" );
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document.write( "* P(R) = P(R|L) * P(L) + P(R|D) * P(D)<BR>\n" );
document.write( "* P(R) = (0.20 * 0.30) + (0.45 * 0.70)<BR>\n" );
document.write( "* P(R) = 0.06 + 0.315<BR>\n" );
document.write( "* P(R) = 0.375\r<BR>\n" );
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document.write( "**Solving for a) P(L|R)**\r<BR>\n" );
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document.write( "* P(L|R) = [P(R|L) * P(L)] / P(R)<BR>\n" );
document.write( "* P(L|R) = (0.20 * 0.30) / 0.375<BR>\n" );
document.write( "* P(L|R) = 0.06 / 0.375<BR>\n" );
document.write( "* P(L|R) = 0.16\r<BR>\n" );
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document.write( "**Solving for b) P(D|R)**\r<BR>\n" );
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document.write( "* P(D|R) = [P(R|D) * P(D)] / P(R)<BR>\n" );
document.write( "* P(D|R) = (0.45 * 0.70) / 0.375<BR>\n" );
document.write( "* P(D|R) = 0.315 / 0.375<BR>\n" );
document.write( "* P(D|R) = 0.84\r<BR>\n" );
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document.write( "**Answers**\r<BR>\n" );
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document.write( "* a) The probability that a student who resigned within two years was hired from a local university is 0.16 or 16%.<BR>\n" );
document.write( "* b) The probability that a student who resigned within two years was hired from a distant university is 0.84 or 84%.<BR>\n" );
document.write( " </SPAN>\n" );
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