document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209744>Question 1209744</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dandelions are studied for their effects on crop production and lawn growth. In one region, the mean number of dandelions per square meter was found to be 9.9.\r<BR>\n" );
document.write( "\n" );
document.write( "Find the probability of no dandelions in an area of 1 m².<BR>\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "Find the probability of at least one dandelion in an area of 1 m².<BR>\n" );
document.write( "P(at least one) = <BR>\n" );
document.write( "0.99995<BR>\n" );
document.write( "Correct\r<BR>\n" );
document.write( "\n" );
document.write( "Find the probability of at most two dandelions in an area of 1 m².<BR>\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850085 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=850085\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You're on the right track using the Poisson distribution! Here's how to calculate the probabilities:\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Probability of no dandelions:**\r<BR>\n" );
document.write( "\n" );
document.write( "* The Poisson distribution formula is: P(x) = (e^-μ * μ^x) / x!<BR>\n" );
document.write( "    * Where:<BR>\n" );
document.write( "        * P(x) is the probability of x events occurring<BR>\n" );
document.write( "        * e is the base of the natural logarithm (~2.71828)<BR>\n" );
document.write( "        * μ is the mean number of events (9.9 dandelions per square meter)<BR>\n" );
document.write( "        * x is the number of events we're interested in (0 dandelions)<BR>\n" );
document.write( "        * x! is the factorial of x (0! = 1)\r<BR>\n" );
document.write( "\n" );
document.write( "* Plugging in the values: P(0) = (e^-9.9 * 9.9^0) / 1 <BR>\n" );
document.write( "* Calculating this gives you approximately 0.0000504\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Probability of at least one dandelion:**\r<BR>\n" );
document.write( "\n" );
document.write( "* This is the complement of having no dandelions. So:<BR>\n" );
document.write( "* P(at least one) = 1 - P(0) <BR>\n" );
document.write( "* P(at least one) = 1 - 0.0000504 <BR>\n" );
document.write( "* P(at least one) ≈ 0.9999496 (which rounds to 0.99995 as you correctly stated)\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Probability of at most two dandelions:**\r<BR>\n" );
document.write( "\n" );
document.write( "* This means we want the probability of having 0, 1, or 2 dandelions. We need to calculate each of these probabilities and add them together:<BR>\n" );
document.write( "    * P(0) = 0.0000504 (calculated above)<BR>\n" );
document.write( "    * P(1) = (e^-9.9 * 9.9^1) / 1! ≈ 0.0004988<BR>\n" );
document.write( "    * P(2) = (e^-9.9 * 9.9^2) / 2! ≈ 0.002469<BR>\n" );
document.write( "* P(at most two) = P(0) + P(1) + P(2)<BR>\n" );
document.write( "* P(at most two) ≈ 0.0000504 + 0.0004988 + 0.002469<BR>\n" );
document.write( "* P(at most two) ≈ 0.0030182\r<BR>\n" );
document.write( "\n" );
document.write( "**Therefore, the probability of at most two dandelions in an area of 1 m² is approximately 0.0030182**<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );