document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1026994>Question 1026994</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a family is planning to have 6 Children, what is the probability of having:<BR>\n" );
document.write( " 1) 5 boys, 1 girl<BR>\n" );
document.write( " 2) 3 girls, 3 boys </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #642282 by </B><A HREF=/tutors/aboutme.mpl?userid=mathmate><B>mathmate(429)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathmate><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=mathmate><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=642282\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Question:<BR>\n" );
document.write( "If a family is planning to have 6 Children, what is the probability of having:<BR>\n" );
document.write( " 1) 5 boys, 1 girl<BR>\n" );
document.write( " 2) 3 girls, 3 boys<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( "Assume equal probabilities of having girls and boys. <BR>\n" );
document.write( "This problem can be solved by drawing a binary probability tree with 64 ultimate branches, and the required answer by counting.<BR>\n" );
document.write( "Alternative to the tree, it could be a table 6 rows (usually columns, but here I put them in rows to make typing easier).<BR>\n" );
document.write( "BGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBGBG<BR>\n" );
document.write( "BBGGBBGGBBGGBBGGBBGGBBGGBBGGBBGGBBGGBBGGBBGGBBGGBBGGBBGGBBGGBBGG<BR>\n" );
document.write( "BBBBGGGGBBBBGGGGBBBBGGGGBBBBGGGGBBBBGGGGBBBBGGGGBBBBGGGGBBBBGGGG<BR>\n" );
document.write( "BBBBBBBBGGGGGGGGBBBBBBBBGGGGGGGGBBBBBBBBGGGGGGGGBBBBBBBBGGGGGGGG<BR>\n" );
document.write( "BBBBBBBBBBBBBBBBGGGGGGGGGGGGGGGGBBBBBBBBBBBBBBBBGGGGGGGGGGGGGGGG<BR>\n" );
document.write( "BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG<BR>\n" );
document.write( "From the above table, seek out the cases where there are 5 boys and 1 girl, or 3 girls and 3 boys.  (Each column represents one case, 64 columns in all).<BR>\n" );
document.write( " <BR>\n" );
document.write( "If you find this method to straining to your eyes, you can always try the binomial probability distribution, which says that<BR>\n" );
document.write( "P(x,n,p)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2Cx%29%2Ap%5Ex%2A%281-p%29%5E%28n-x%29 ALIGN=MIDDLE  ALT=\"C%28n%2Cx%29%2Ap%5Ex%2A%281-p%29%5E%28n-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where<BR>\n" );
document.write( "x is the number of boys<BR>\n" );
document.write( "n is the total number of children (6)<BR>\n" );
document.write( "p is the probability of having a boy (0.5)<BR>\n" );
document.write( "and<BR>\n" );
document.write( "C(n,x) is the binomial coefficient of taking x objects out of n, and equals<BR>\n" );
document.write( "n!/(x!(n-x)!)<BR>\n" );
document.write( " <BR>\n" );
document.write( "Hence<BR>\n" );
document.write( "(1) P(5 boys, 6,0.5) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%286%2C5%29%2A%280.5%5E5%29%2A%280.5%5E1%29+=+6%2A%281%2F32%29%281%2F2%29=6%2F64=3%2F32 ALIGN=MIDDLE  ALT=\"C%286%2C5%29%2A%280.5%5E5%29%2A%280.5%5E1%29+=+6%2A%281%2F32%29%281%2F2%29=6%2F64=3%2F32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(2) P(3 boys, 6,0.5) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%286%2C3%29%2A%280.5%5E3%29%2A%280.5%5E3%29+=+20%281%2F8%29%281%2F8%29=20%2F64=5%2F16 ALIGN=MIDDLE  ALT=\"C%286%2C3%29%2A%280.5%5E3%29%2A%280.5%5E3%29+=+20%281%2F8%29%281%2F8%29=20%2F64=5%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );