document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172793>Question 1172793</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ten is the average number of oil tankers arriving each day at a certain port city. The facilities at the port can handle at most 10 tankers per day. What is the probability that on a given day\r<BR>\n" );
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document.write( "(a) Tankers have to be turned away?<BR>\n" );
document.write( "(b) Tankers have to be facilitated?<BR>\n" );
document.write( "(c) No truck arrived at the port?<BR>\n" );
document.write( "(d) Exactly 8 trucks have to be facilitated? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850802 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=850802\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem using the Poisson distribution.\r<BR>\n" );
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document.write( "**Given:**\r<BR>\n" );
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document.write( "* Average number of tankers arriving per day (λ) = 10<BR>\n" );
document.write( "* Port capacity = 10 tankers per day\r<BR>\n" );
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document.write( "**a. Probability Tankers Have to Be Turned Away**\r<BR>\n" );
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document.write( "* Tankers are turned away if more than 10 arrive.<BR>\n" );
document.write( "* We need to find P(X > 10), where X is the number of tankers arriving.<BR>\n" );
document.write( "* P(X > 10) = 1 - P(X ≤ 10)<BR>\n" );
document.write( "* We can use the Poisson distribution formula: P(X = k) = (e^(-λ) * λ^k) / k!<BR>\n" );
document.write( "* P(X ≤ 10) = Σ [ (e^(-10) * 10^k) / k! ] for k = 0 to 10<BR>\n" );
document.write( "* Using a calculator or statistical software:<BR>\n" );
document.write( "    * P(X ≤ 10) ≈ 0.5830<BR>\n" );
document.write( "* P(X > 10) = 1 - 0.5830 ≈ 0.4170\r<BR>\n" );
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document.write( "**b. Probability Tankers Have to Be Facilitated**\r<BR>\n" );
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document.write( "* Tankers are facilitated if 10 or fewer arrive.<BR>\n" );
document.write( "* This is P(X ≤ 10).<BR>\n" );
document.write( "* From part (a), we already found P(X ≤ 10) ≈ 0.5830\r<BR>\n" );
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document.write( "**c. Probability No Truck Arrived at the Port**\r<BR>\n" );
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document.write( "* We need to find P(X = 0).<BR>\n" );
document.write( "* P(X = 0) = (e^(-10) * 10^0) / 0!<BR>\n" );
document.write( "* P(X = 0) = e^(-10) * 1 / 1<BR>\n" );
document.write( "* P(X = 0) ≈ 0.0000454\r<BR>\n" );
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document.write( "**d. Probability Exactly 8 Trucks Have to Be Facilitated**\r<BR>\n" );
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document.write( "* We need to find P(X = 8).<BR>\n" );
document.write( "* P(X = 8) = (e^(-10) * 10^8) / 8!<BR>\n" );
document.write( "* P(X = 8) = (e^(-10) * 100000000) / 40320<BR>\n" );
document.write( "* P(X = 8) ≈ 0.1126\r<BR>\n" );
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document.write( "**Summary:**\r<BR>\n" );
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document.write( "* **(a) Probability tankers have to be turned away:** ≈ 0.4170<BR>\n" );
document.write( "* **(b) Probability tankers have to be facilitated:** ≈ 0.5830<BR>\n" );
document.write( "* **(c) Probability no truck arrived at the port:** ≈ 0.0000454<BR>\n" );
document.write( "* **(d) Probability exactly 8 trucks have to be facilitated:** ≈ 0.1126<BR>\n" );
document.write( " </SPAN>\n" );
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