document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1023496>Question 1023496</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A couple is planning to have 4 children. Assume boys and girls are equally likely and that the gender of any child is not influenced by gender of any other child. Find the probability of getting exactly 2 boys and 2 girls... Find the probability that the 4 children are all boys. </SPAN>\n" );
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document.write( "Question:<BR>\n" );
document.write( "A couple is planning to have 4 children. Assume boys and girls are equally likely and that the gender of any child is not influenced by gender of any other child. Find the probability of getting exactly 2 boys and 2 girls... Find the probability that the 4 children are all boys.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( "This is a typical application of the tree diagram.<BR>\n" );
document.write( "If you create a tree diagram with 4 stages, you will end up with 16 outcomes.<BR>\n" );
document.write( "If you wish, you can replace the tree diagram with a table<BR>\n" );
document.write( "The sixteen outcomes are (by columns)<BR>\n" );
document.write( "BBBBBBBBGGGGGGGG<BR>\n" );
document.write( "BBBBGGGGBBBBGGGG<BR>\n" );
document.write( "BBGGBBGGBBGGBBGG<BR>\n" );
document.write( "BGBGBGBGBGBGBGBG<BR>\n" );
document.write( "So you can sort out, for example, there is only one column that reads BBBB, so four boys is 1/16<BR>\n" );
document.write( "There are columns that read BBBG, BBGB, BGBB, GBBB, so 4/16 is the probability for one girl, (or three boys) etc.<BR>\n" );
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document.write( "Another way is to use the binomial distribution, which says<BR>\n" );
document.write( "P(k;n;p)=C(n,k)*p^k*(1-p)^(n-k)<BR>\n" );
document.write( "where<BR>\n" );
document.write( "k=number of boys in the outcome<BR>\n" );
document.write( "n=total number of children<BR>\n" );
document.write( "p=probability of success, i.e. having a boy at a particular step (1/2) <BR>\n" );
document.write( "C(n,k)=binary coefficient = n!/(k!(n-k!))<BR>\n" );
document.write( " <BR>\n" );
document.write( "For example, the probability of having 2 boys (out of 4 children)<BR>\n" );
document.write( "=P(2;4;1/2)=C(4,2)*(1/2)^2*(1/2)^2=4!/(2!2!)*1/4*1/4=6/16=3/8. </SPAN>\n" );
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