document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199984>Question 1199984</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If 4% of the population carries a certain<BR>\n" );
document.write( "genetic trait, hnd the probability that in a sample<BR>\n" );
document.write( "of 100 people, there are exactly 8 people who have<BR>\n" );
document.write( "the trait. Assume the distribution is approximately<BR>\n" );
document.write( "Poisson. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848135 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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=848135\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Determine the Average Number of Occurrences (λ)**\r<BR>\n" );
document.write( "\n" );
document.write( "* **λ = n * p** <BR>\n" );
document.write( "    * Where:<BR>\n" );
document.write( "        * n = sample size (100 people)<BR>\n" );
document.write( "        * p = probability of the trait (0.04) <BR>\n" );
document.write( "    * λ = 100 * 0.04 = 4\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Use the Poisson Probability Mass Function**\r<BR>\n" );
document.write( "\n" );
document.write( "* The probability of observing exactly *k* events in a Poisson distribution is given by:\r<BR>\n" );
document.write( "\n" );
document.write( "   * P(X = k) = (e^(-λ) * λ^k) / k! \r<BR>\n" );
document.write( "\n" );
document.write( "   * Where:<BR>\n" );
document.write( "      * X is the random variable representing the number of people with the trait<BR>\n" );
document.write( "      * λ is the average number of occurrences (4)<BR>\n" );
document.write( "      * k is the desired number of occurrences (8)<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 (k! = k * (k-1) * (k-2) * ... * 1)\r<BR>\n" );
document.write( "\n" );
document.write( "* **Calculate P(X = 8):**\r<BR>\n" );
document.write( "\n" );
document.write( "   * P(X = 8) = (e^(-4) * 4^8) / 8! <BR>\n" );
document.write( "   * P(X = 8) ≈ (0.0183 * 65536) / 40320 <BR>\n" );
document.write( "   * P(X = 8) ≈ 0.0298\r<BR>\n" );
document.write( "\n" );
document.write( "**Therefore, the probability that in a sample of 100 people, exactly 8 people have the genetic trait is approximately 0.0298 or 2.98%.**<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );