document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191373>Question 1191373</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find each Poisson probability, using a mean arrival rate of 10 arrivals per hour.\r<BR>\n" );
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document.write( "(c) Fewer than five arrivals. (Round your answer to 4 decimal places.)<BR>\n" );
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document.write( " Poisson probability: \r<BR>\n" );
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document.write( "(d) At least 11 arrivals. (Round your answer to 4 decimal places.)<BR>\n" );
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document.write( " Poisson probability:  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849173 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=849173\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the Poisson probabilities:\r<BR>\n" );
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document.write( "**Understanding the Poisson Distribution**\r<BR>\n" );
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document.write( "The Poisson distribution is used to model the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known average rate and independently of the time since the last event. The formula is:\r<BR>\n" );
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document.write( "P(x) = (e^(-λ) * λ^x) / x!\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* P(x) is the probability of x events occurring<BR>\n" );
document.write( "* e is the base of the natural logarithm (approximately 2.71828)<BR>\n" );
document.write( "* λ is the average number of events (arrivals) per interval (10 in this case)<BR>\n" );
document.write( "* x is the number of events we're interested in<BR>\n" );
document.write( "* x! is the factorial of x\r<BR>\n" );
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document.write( "**c) Fewer than five arrivals (x < 5):**\r<BR>\n" );
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document.write( "We need to calculate P(0) + P(1) + P(2) + P(3) + P(4).\r<BR>\n" );
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document.write( "* P(0) = (e^-10 * 10^0) / 0! ≈ 0.000045<BR>\n" );
document.write( "* P(1) = (e^-10 * 10^1) / 1! ≈ 0.000454<BR>\n" );
document.write( "* P(2) = (e^-10 * 10^2) / 2! ≈ 0.002270<BR>\n" );
document.write( "* P(3) = (e^-10 * 10^3) / 3! ≈ 0.007567<BR>\n" );
document.write( "* P(4) = (e^-10 * 10^4) / 4! ≈ 0.018918\r<BR>\n" );
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document.write( "P(x < 5) = P(0) + P(1) + P(2) + P(3) + P(4) ≈ 0.000045 + 0.000454 + 0.002270 + 0.007567 + 0.018918 ≈ 0.029254\r<BR>\n" );
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document.write( "Rounded to 4 decimal places, the probability is 0.0293.\r<BR>\n" );
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document.write( "**d) At least 11 arrivals (x ≥ 11):**\r<BR>\n" );
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document.write( "It's easier to calculate the complement and subtract from 1.  That is:\r<BR>\n" );
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document.write( "P(x ≥ 11) = 1 - P(x < 11) = 1 - [P(0) + P(1) + ... + P(10)]\r<BR>\n" );
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document.write( "Calculating each individual probability from P(0) to P(10) and summing them can be tedious.  It's best to use a calculator or statistical software that has built-in Poisson cumulative distribution functions (CDF).<BR>\n" );
document.write( "Using a calculator or software, you'll find that P(x < 11) ≈ 0.5830.\r<BR>\n" );
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document.write( "Therefore, P(x ≥ 11) = 1 - 0.5830 = 0.4170\r<BR>\n" );
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document.write( "Rounded to 4 decimal places, the probability is 0.4170.<BR>\n" );
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