document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1011694>Question 1011694</A>: <I><SPAN STYLE=\"word-break: break-all;\"> To estimate the number of fish in a small lake, researchers caught 200 fish, tagged them, and returned them to the lake. The next day they caught 100 fish. Of those caught, 25 had a tag. If there were 800 fish in the lake, what is the probability that 25 of the 100 fish in the second catch would have been tagged? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #627467 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=627467\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you tag 200 fish from the lake.<BR>\n" );
document.write( "assuming that there are 800 fish in the lake, what is the probability that a sample of 100 fish taken from the lake would contain exactly 25 tagged fish.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if you tag 200 fish from the lake, and there are 800 fish in the lake, then the proportion of tagged fish to total fish is .25.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you subsequently catch 100 fish and want to know the probability that exactly 25 of those fish will be tagged.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this is a binomial distribution.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formula for a binomial distribution is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "p(x) = c(n,x) * p^x * q^(n-x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = 25<BR>\n" );
document.write( "n = 100<BR>\n" );
document.write( "p = .25<BR>\n" );
document.write( "q = 1 - .25 = .75\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formula becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "p(25) = c(100,25) * .25^25 * .75^75\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if your calculator can handle it, you would get p(25) = .0917996918 which we'll round off to .0918.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you get p(25) = .0918.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "c(n,x) is the combination formula of n! / (x! * (n-x)!)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "it tells you how many possible sets you can get of x elements each out of a total of n elements when order is not important.  sets that contain the same elements but in a different order are considered to be the same set and are only counted once.   this is different from a permutation where sets that have the same elements but in a different order are counted as different sets.<BR>\n" );
document.write( "for example:<BR>\n" );
document.write( "abc and cba are counted as 1 set by the combination formula.<BR>\n" );
document.write( "abc and cba are counted as 2 sets by the permutation formula.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if your calculator can't handle it, then you would use the normal approximation to the binomial.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in that case, you would do the following.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "m = n*p = 100*.25 = 25<BR>\n" );
document.write( "s = sqrt(n*p*q) = sqrt(100*.25*.75) = 4.330127019\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "m1 = m - .5 = 25 - .5 = 24.5<BR>\n" );
document.write( "m2 = m + .5 = 25 + .5 = 25.5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z1 = (24.5 - 25) / 4.330127019 = -.154700538<BR>\n" );
document.write( "z2 = (25.5 - 25) / 4.330127019 = .154700538\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you would then use your z-score calculator to find the area under the distribution curve between a z-factor of -.154700538 and .154700538.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the result will be that the area under the distribution curve = .0919275415 which you would round off to .0919.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "with the binomial distribution formula, you got .0918.<BR>\n" );
document.write( "with the normal approximation to the binomial distribution formula, you got .0919.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "they're very close to each other, so either one would have sufficed.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "assuming the binomial distribution formula is the more accurate, the normal approximation of .0919 would be off .0001 from the binomial approximation of .0918 which is about a tenth of a percent off.<BR>\n" );
document.write( "that's pretty good.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the first thing that you needed to do to solve this problem was to understand that it is a binomial distribution type of problem.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the second thing that you needed to know is that you could use the normal approximation to the binomial if you had to.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here's some references you might like to look at for further information.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.stattrek.com/probability-distributions/binomial.aspx\" target = \"_blank\">http://www.stattrek.com/probability-distributions/binomial.aspx</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.mathsisfun.com/data/binomial-distribution.html\" target = \"_blank\">http://www.mathsisfun.com/data/binomial-distribution.html</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"http://regentsprep.org/Regents/math/algtrig/ATS7/BLesson3.htm\" target = \"_blank\">http://regentsprep.org/Regents/math/algtrig/ATS7/BLesson3.htm</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"http://onlinestatbook.com/2/normal_distribution/normal_approx.html\" target = \"_blank\">http://onlinestatbook.com/2/normal_distribution/normal_approx.html</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "there are plenty more on the web.<BR>\n" );
document.write( "just do a search for (binomial distribution) or (normal approximation to the binomial) and you'll get tons of them.\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );