document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=200135>Question 200135</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I got this one wrong:\r<BR>\n" );
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document.write( "A couple plan to have exactly four children. \r<BR>\n" );
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document.write( "(a)	Construct a tree diagram and list the sample space.<BR>\n" );
document.write( "(b)	Find the probability that the family has at least three boys.   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #150448 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=150448\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a)\r<BR>\n" );
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document.write( "Let's build the tree:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/work/w5a39a-2.png\">\r<BR>\n" );
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document.write( "Using the tree, we get the sample space:\r<BR>\n" );
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document.write( "{b,b,b,b}<BR>\n" );
document.write( "{b,b,b,g}<BR>\n" );
document.write( "{b,b,g,b}<BR>\n" );
document.write( "{b,b,g,g}<BR>\n" );
document.write( "{b,g,b,b}<BR>\n" );
document.write( "{b,g,b,g}<BR>\n" );
document.write( "{b,g,g,b}<BR>\n" );
document.write( "{b,g,g,g}\r<BR>\n" );
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document.write( "{g,b,b,b}<BR>\n" );
document.write( "{g,b,b,g}<BR>\n" );
document.write( "{g,b,g,b}<BR>\n" );
document.write( "{g,b,g,g}<BR>\n" );
document.write( "{g,g,b,b}<BR>\n" );
document.write( "{g,g,b,g}<BR>\n" );
document.write( "{g,g,g,b}<BR>\n" );
document.write( "{g,g,g,g}\r<BR>\n" );
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document.write( "So for instance {b,g,g,b} means that the couple had a boy, girl, girl, and then a boy (in that order).\r<BR>\n" );
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document.write( "Note: recall, the sample space is the set of ALL possible outcomes.\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "Since we want to know the chances of the couple having \"at least three boys\", this means they want to know the chances of having 3 boys OR 4 boys (since <i>at least</i> means that figure or more).\r<BR>\n" );
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document.write( "Looking back at the list of all possible outcomes (ie the sample space) from part a), we see that we have the combinations for 3 boys:\r<BR>\n" );
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document.write( "{b,b,b,g}, {b,b,g,b}, {b,g,b,b}, and {g,b,b,b}\r<BR>\n" );
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document.write( "So there are 4 cases where the couple would have 3 boys. \r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(\text{3 boys ONLY})=\frac{\text{Number of Cases with 3 boys only}}{\text{Total number in sample space}}=\frac{4}{16}=\frac{1}{4}\"> \r<BR>\n" );
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document.write( "Also, since there is only ONE way to have 4 boys (of a total of 4 children), this means that<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(\text{\text{4 boys ONLY}})=\frac{\text{Number of Cases with 4 boys only}}{\text{Total number in sample space}}=\frac{1}{16}\">\r<BR>\n" );
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document.write( "Now simply add the two probabilities to find the chances of either one occurring:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(\text{AT LEAST 3 boys})=P(\text{3 boys OR 4 boys})\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(\text{AT LEAST 3 boys})=P(\text{3 boys ONLY})+P(\text{4 boys ONLY})\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(\text{AT LEAST 3 boys})=\frac{\text{Number of Cases with 3 boys only}}{\text{Total number in sample space}}+\frac{\text{Number of Cases with 4 boys only}}{\text{Total number in sample space}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(\text{AT LEAST 3 boys})=\frac{4}{16}+\frac{1}{16}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(\text{AT LEAST 3 boys})=\frac{4+1}{16}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(\text{AT LEAST 3 boys})=\frac{5}{16}\">\r<BR>\n" );
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document.write( "So the probability of the couple having AT LEAST 3 boys is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F16 ALIGN=MIDDLE  ALT=\"5%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is 0.3125 in decimal form which gives a 31.25% chance.<BR>\n" );
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