document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=333310>Question 333310</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.	Suppose a survey of 890 families with 5 children was done<BR>\n" );
document.write( "a)	What is the probability of observing children to be all boys or all girls </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #238828 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=238828\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It depends.  What are your assumptions regarding the probability of conception of a boy vs. a girl?  If you presume 50/50, which is totally false, then you would proceed thus:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The probability of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> successes in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> trials where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\"> is the probability of success on any given trial is given by:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_n(k,p)\ =\ \left(n\cr k\right\)p^k\left(1\,-\,p\right)^{n\,-\,k}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(n\cr k\right\)\"> is the number of combinations of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> things taken <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> at a time and is calculated by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{n!}{k!(n\,-\,k)!}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To calculate the probability of all boys in 5, you would use:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_5(5,0.5)\ =\ \left(5\cr 5\right\)(0.5)^5(0.5)^0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Fortunately, this reduces to a very simple calculation because <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(n\cr n\right\)\ =\ 1\ \forall\ n\ >\ 0\ \in\ \mathbb{Z}\">.  And <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^0\ =\ 1\ \forall\ x\ \in\ \mathbb{R}\">.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the probability is 1/2, the probability of 5 girls is the same, so the probability of either is just two times either.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If you want to assume the empirical probability of a boy being 105/205 or approximately 0.512, then you need to do:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_5(5,0.512)\ +\ P_5(5,0.488)\ =\ (0.512)^5\ +\ (0.488)^5\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Once you decide on which probability to use, multiply by 890 to find the expected number.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg\"><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );