document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169620>Question 1169620</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Mary can take on three possible routes from her home to her office in the morning.<BR>\n" );
document.write( "The percentage of days on routes A, B and C to her office were 45%, 30% and<BR>\n" );
document.write( "25% respectively. The traveling time on each of these three routes is normally<BR>\n" );
document.write( "distributed with mean and standard deviation shown in the following table:<BR>\n" );
document.write( "                    Route A      Route B     Route C<BR>\n" );
document.write( "Mean               30 minutes  35 minutes  28 minutes<BR>\n" );
document.write( "Standard deviation 10 minutes  8 minutes  12 minutes\r<BR>\n" );
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document.write( "Mary left her home at 8:28 a.m. and arrived at her office before 9:00 a.m.,<BR>\n" );
document.write( "calculate the probability that she took route B to work.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851309 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=851309\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve 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( "We need to calculate the probability that Mary took route B, given that she arrived at her office before 9:00 a.m.\r<BR>\n" );
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document.write( "**Given Information**\r<BR>\n" );
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document.write( "* Routes: A, B, C<BR>\n" );
document.write( "* Probabilities: P(A) = 0.45, P(B) = 0.30, P(C) = 0.25<BR>\n" );
document.write( "* Arrival time: Before 9:00 a.m. (32 minutes after leaving home)<BR>\n" );
document.write( "* Mean and standard deviation of travel times:<BR>\n" );
document.write( "    * Route A: μA = 30, σA = 10<BR>\n" );
document.write( "    * Route B: μB = 35, σB = 8<BR>\n" );
document.write( "    * Route C: μC = 28, σC = 12\r<BR>\n" );
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document.write( "**Solution**\r<BR>\n" );
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document.write( "1.  **Define Events:**\r<BR>\n" );
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document.write( "    * A: Mary took route A.<BR>\n" );
document.write( "    * B: Mary took route B.<BR>\n" );
document.write( "    * C: Mary took route C.<BR>\n" );
document.write( "    * E: Mary arrived before 9:00 a.m. (travel time ≤ 32 minutes)\r<BR>\n" );
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document.write( "2.  **Calculate P(E|A), P(E|B), P(E|C):**\r<BR>\n" );
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document.write( "    * **P(E|A):** Probability of arriving before 9:00 a.m. given she took route A.<BR>\n" );
document.write( "        * zA = (32 - 30) / 10 = 2 / 10 = 0.2<BR>\n" );
document.write( "        * P(E|A) = P(ZA ≤ 0.2) ≈ 0.5793 (using a z-table or calculator)<BR>\n" );
document.write( "    * **P(E|B):** Probability of arriving before 9:00 a.m. given she took route B.<BR>\n" );
document.write( "        * zB = (32 - 35) / 8 = -3 / 8 = -0.375<BR>\n" );
document.write( "        * P(E|B) = P(ZB ≤ -0.375) ≈ 0.3538 (using a z-table or calculator)<BR>\n" );
document.write( "    * **P(E|C):** Probability of arriving before 9:00 a.m. given she took route C.<BR>\n" );
document.write( "        * zC = (32 - 28) / 12 = 4 / 12 = 1/3 ≈ 0.3333<BR>\n" );
document.write( "        * P(E|C) = P(ZC ≤ 0.3333) ≈ 0.6305 (using a z-table or calculator)\r<BR>\n" );
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document.write( "3.  **Calculate P(E):**\r<BR>\n" );
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document.write( "    * P(E) = P(E|A)P(A) + P(E|B)P(B) + P(E|C)P(C)<BR>\n" );
document.write( "    * P(E) = (0.5793 * 0.45) + (0.3538 * 0.30) + (0.6305 * 0.25)<BR>\n" );
document.write( "    * P(E) = 0.260685 + 0.10614 + 0.157625<BR>\n" );
document.write( "    * P(E) ≈ 0.52445\r<BR>\n" );
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document.write( "4.  **Apply Bayes' Theorem:**\r<BR>\n" );
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document.write( "    * We want to find P(B|E), the probability that she took route B given she arrived before 9:00 a.m.<BR>\n" );
document.write( "    * Bayes' Theorem: P(B|E) = (P(E|B) * P(B)) / P(E)<BR>\n" );
document.write( "    * P(B|E) = (0.3538 * 0.30) / 0.52445<BR>\n" );
document.write( "    * P(B|E) = 0.10614 / 0.52445<BR>\n" );
document.write( "    * P(B|E) ≈ 0.2024\r<BR>\n" );
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document.write( "**Final Answer**\r<BR>\n" );
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document.write( "The probability that Mary took route B to work is approximately 0.2024.<BR>\n" );
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