document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=250486>Question 250486</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A couple decided to have 4 children. <BR>\n" );
document.write( "What is the probability that they will have at least two boys?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #182372 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=182372\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the possible combinations of boys and girls are:\r<BR>\n" );
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document.write( "bbbb<BR>\n" );
document.write( "bbbg<BR>\n" );
document.write( "bbgg<BR>\n" );
document.write( "bggg<BR>\n" );
document.write( "gggg\r<BR>\n" );
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document.write( "all boys can occur in 1 way so p(4b) = 1 * .5^4 = .0625<BR>\n" );
document.write( "3 boys 1 girl can occur in 4 ways so p(3b1g) = 4 * .5^4 = .25<BR>\n" );
document.write( "2 boys 2 girls can occur in 6 ways so p(2b2g) = 6 * .5^4 = .375<BR>\n" );
document.write( "1 boy 2 girls can occur in 4 ways so p(1b3g) = 4 * .5^4 = .25<BR>\n" );
document.write( "all girls can occur in 1 way so p(4g) = 1 * .5^4 = .0625\r<BR>\n" );
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document.write( "sum of all probabilities should be equal to 1.\r<BR>\n" );
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document.write( "2 * .0625 + 2 * .25 + .375 = 1 so we're good there.\r<BR>\n" );
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document.write( "probability of at least 2 boys equals p(2b2g) + p(3b1g) + p(4b) = .375 + .25 + .0625 = .6875 = 68.75%\r<BR>\n" );
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document.write( "the number of ways you can get all boys is 1 as shown below:\r<BR>\n" );
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document.write( "bbbb\r<BR>\n" );
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document.write( "the number of ways you can get 3 boys and 1 girls is 4 as shown below:\r<BR>\n" );
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document.write( "bbbg<BR>\n" );
document.write( "bbgb<BR>\n" );
document.write( "bgbb<BR>\n" );
document.write( "gbbb\r<BR>\n" );
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document.write( "the number of ways you can get 2 boys and 1 girl is 6 as shown below:\r<BR>\n" );
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document.write( "bbgg<BR>\n" );
document.write( "bggb<BR>\n" );
document.write( "ggbb<BR>\n" );
document.write( "gbbg<BR>\n" );
document.write( "bgbg<BR>\n" );
document.write( "gbgb\r<BR>\n" );
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document.write( "since the order in which the children arrive has something to do with the probability of them arriving, these would be permutations.\r<BR>\n" );
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document.write( "since the probability of having a boy or a girl was the same (.5 in each case), the probability part of the equations was always the same (.5^4) regardless of whether it was 1 boy, 2 boys, 3 boys, or 4 boys.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "p(bbbb) = .5*.5*.5*.5 = .5^4<BR>\n" );
document.write( "p(gggg) = .5*.5*.5*.5 = .5^4<BR>\n" );
document.write( "p(bbbg) = .5*.5*.5*.5 = .5^4<BR>\n" );
document.write( "p(bbgg) = .5*.5*.5*.5 = .5^4\r<BR>\n" );
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document.write( "if the probability was different, i.e. p(b) = .75, p(g) = .25, then the probability for each of the permutations would have been different.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "p(bbbb) = .25*.25*.25*.25 = .25^4<BR>\n" );
document.write( "p(gggg) = .75*.75*.75*.75 = .75^4<BR>\n" );
document.write( "p(bbbg) = .25*.25*.25*.75 = .25^3*.75<BR>\n" );
document.write( "p(bbgg) = .25*.25*.75*.75 = .25^2*.75^2\r<BR>\n" );
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document.write( " </SPAN>\n" );
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