document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200444>Question 1200444</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport. The mean arrival rate is 11 passengers per minute.\r<BR>\n" );
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document.write( "a. Compute the probability of no arrivals in a one-minute period (to 6 decimals). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848094 by </B><A HREF=/tutors/aboutme.mpl?userid=GingerAle><B>GingerAle(43)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=GingerAle><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848094\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Identify the Distribution**\r<BR>\n" );
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document.write( "* Since passengers arrive randomly and independently at a constant average rate, this situation can be modeled by a **Poisson distribution**.\r<BR>\n" );
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document.write( "**2. Poisson Probability Formula**\r<BR>\n" );
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document.write( "* The probability of observing *k* events in a given time interval, given the average rate (λ) of events occurring in that interval, is:\r<BR>\n" );
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document.write( "  * P(X = k) = (λ^k * e^(-λ)) / k! \r<BR>\n" );
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document.write( "  where:<BR>\n" );
document.write( "    * X is the number of events (passengers)<BR>\n" );
document.write( "    * λ is the average arrival rate (11 passengers/minute)<BR>\n" );
document.write( "    * k is the desired number of events (0 passengers in this case)<BR>\n" );
document.write( "    * e is the base of the natural logarithm (approximately 2.71828)<BR>\n" );
document.write( "    * k! is the factorial of k (0! = 1)\r<BR>\n" );
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document.write( "**3. Calculate Probability of No Arrivals**\r<BR>\n" );
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document.write( "* P(X = 0) = (11^0 * e^(-11)) / 0! <BR>\n" );
document.write( "* P(X = 0) = (1 * e^(-11)) / 1<BR>\n" );
document.write( "* P(X = 0) = e^(-11) <BR>\n" );
document.write( "* P(X = 0) ≈ 0.0000000016 \r<BR>\n" );
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document.write( "**Therefore, the probability of no arrivals in a one-minute period is approximately 0.0000000016.**<BR>\n" );
document.write( " </SPAN>\n" );
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