document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177981>Question 1177981</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The University of America runs two bus lines on campus: red and green. The red line serves north<BR>\n" );
document.write( "campus and the green line serves south campus with a transfer station linking the two lines. Green<BR>\n" );
document.write( "buses arrive randomly (according to a Poisson distribution) at the transfer station every 10 minutes.<BR>\n" );
document.write( "Red buses also arrive randomly every 7 minutes.<BR>\n" );
document.write( "(a) What is the probability that 2 buses will stop at the station during a 5 minute interval?<BR>\n" );
document.write( "(b) A student whose dormitory is located next to the station has a class in 10 minutes. Either bus will<BR>\n" );
document.write( "take the student to the classroom building. The ride takes 5 minutes, after which the student will walk<BR>\n" );
document.write( "for around 3 minutes to reach the class room. What is the probability that the student will make it to<BR>\n" );
document.write( "class on time? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850390 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=850390\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**Understanding the Problem**\r<BR>\n" );
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document.write( "* **Green Buses:** Poisson distribution, arrival every 10 minutes (λ_green = 10 minutes)<BR>\n" );
document.write( "* **Red Buses:** Poisson distribution, arrival every 7 minutes (λ_red = 7 minutes)\r<BR>\n" );
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document.write( "**(a) Probability of 2 Buses in a 5-Minute Interval**\r<BR>\n" );
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document.write( "1.  **Calculate Arrival Rates for 5 Minutes**\r<BR>\n" );
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document.write( "    * Green buses: λ_green(5) = (5 minutes / 10 minutes) = 0.5 buses<BR>\n" );
document.write( "    * Red buses: λ_red(5) = (5 minutes / 7 minutes) ≈ 0.7143 buses\r<BR>\n" );
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document.write( "2.  **Calculate Total Arrival Rate**\r<BR>\n" );
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document.write( "    * Total arrival rate (λ_total) = λ_green(5) + λ_red(5) = 0.5 + 0.7143 = 1.2143 buses\r<BR>\n" );
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document.write( "3.  **Use Poisson Distribution Formula**\r<BR>\n" );
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document.write( "    * P(X = k) = (e^(-λ) * λ^k) / k!\r<BR>\n" );
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document.write( "    We want to find P(X = 2), where λ = 1.2143.\r<BR>\n" );
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document.write( "    * P(X = 2) = (e^(-1.2143) * 1.2143^2) / 2!<BR>\n" );
document.write( "    * P(X = 2) = (0.2969 * 1.4745) / 2<BR>\n" );
document.write( "    * P(X = 2) ≈ 0.4378 / 2<BR>\n" );
document.write( "    * P(X = 2) ≈ 0.2189\r<BR>\n" );
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document.write( "    Therefore, the probability that 2 buses will stop at the station during a 5-minute interval is approximately 0.2189.\r<BR>\n" );
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document.write( "**(b) Probability of Making it to Class on Time**\r<BR>\n" );
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document.write( "1.  **Determine Arrival Rates for 10 Minutes**\r<BR>\n" );
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document.write( "    * Green buses: λ_green(10) = (10 minutes / 10 minutes) = 1 bus<BR>\n" );
document.write( "    * Red buses: λ_red(10) = (10 minutes / 7 minutes) ≈ 1.4286 buses\r<BR>\n" );
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document.write( "2.  **Calculate Probabilities of No Buses Arriving in 10 Minutes**\r<BR>\n" );
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document.write( "    * P(Green = 0) = (e^(-1) * 1^0) / 0! = e^(-1) ≈ 0.3679<BR>\n" );
document.write( "    * P(Red = 0) = (e^(-1.4286) * 1.4286^0) / 0! = e^(-1.4286) ≈ 0.2394\r<BR>\n" );
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document.write( "3.  **Calculate Probability of No Buses Arriving (Both Lines)**\r<BR>\n" );
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document.write( "    * P(Green = 0 and Red = 0) = P(Green = 0) * P(Red = 0) = 0.3679 * 0.2394 ≈ 0.0881\r<BR>\n" );
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document.write( "4.  **Calculate Probability of at Least One Bus Arriving**\r<BR>\n" );
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document.write( "    * P(At least one bus) = 1 - P(Green = 0 and Red = 0) = 1 - 0.0881 = 0.9119\r<BR>\n" );
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document.write( "5.  **Conclusion**\r<BR>\n" );
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document.write( "    If at least one bus arrives within the 10 minute period, the student will have 5 minutes of travel time and 3 minutes of walking time, which will get them to class on time.\r<BR>\n" );
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document.write( "    Therefore, the probability that the student will make it to class on time is approximately 0.9119.\r<BR>\n" );
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document.write( "**Results**\r<BR>\n" );
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document.write( "(a) The probability that 2 buses will stop at the station during a 5-minute interval is approximately 0.2189.<BR>\n" );
document.write( "(b) The probability that the student will make it to class on time is approximately 0.9119.<BR>\n" );
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