document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1026251>Question 1026251</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Assume that the probability of the birth of a child of a particular sex is 50%. In a family with 4 children, what is the probability that:<BR>\n" );
document.write( "a). all children are boys<BR>\n" );
document.write( "b). all children are the same sex<BR>\n" );
document.write( "c).there is at least one boy\r<BR>\n" );
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document.write( "I tried dividing out 50%, but I'm  just more confused. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #641485 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=641485\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's list out the sample space. This is the entire set of all possibilities\r<BR>\n" );
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document.write( "Group 1:<BR>\n" );
document.write( "BBBB\r<BR>\n" );
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document.write( "Group 2:<BR>\n" );
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( "Group 3:<BR>\n" );
document.write( "BBGG<BR>\n" );
document.write( "BGBG<BR>\n" );
document.write( "BGGB<BR>\n" );
document.write( "GBGB<BR>\n" );
document.write( "GGBB<BR>\n" );
document.write( "GBBG\r<BR>\n" );
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document.write( "Group 4:<BR>\n" );
document.write( "GGGB<BR>\n" );
document.write( "GGBG<BR>\n" );
document.write( "GBGG<BR>\n" );
document.write( "BGGG\r<BR>\n" );
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document.write( "Group 5:<BR>\n" );
document.write( "GGGG\r<BR>\n" );
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document.write( "B represents a boy. G represents a girl. <BR>\n" );
document.write( "Group 1 shows that there is only one way to have all 4 boys and no girls.<BR>\n" );
document.write( "Group 2 shows there are 4 ways to have three boys and one girl. <BR>\n" );
document.write( "Group 3 has 6 possibilities of having 2 boys and 2 girls.<BR>\n" );
document.write( "Group 4 is the opposite of group 2. There are 4 ways to have 3 girls and 1 boy<BR>\n" );
document.write( "Group 5 is the opposite of group 1. There is only one way to have 4 girls and no boys\r<BR>\n" );
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document.write( "In total, there are 1+4+6+4+1 = 16 ways to have 4 kids\r<BR>\n" );
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document.write( "Things to note:<BR>\n" );
document.write( "a) There are 4 slots and 2 choices per slot. It's no coincidence that 2^4 = 2*2*2*2 = 16<BR>\n" );
document.write( "b) The sequence 1,4,6,4,1 is found in pascals triangle\r<BR>\n" );
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document.write( "Now onto the problem\r<BR>\n" );
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document.write( "a). all children are boys\r<BR>\n" );
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document.write( "As shown in the sample space above, there is only one way to have all 4 boys which is BBBB. The order of the boys doesn't really matter. This is out of 16 possible ways to have 4 kids. \r<BR>\n" );
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document.write( "The probability is 1/16 = 0.0625 = 6.25%\r<BR>\n" );
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document.write( "b). all children are the same sex\r<BR>\n" );
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document.write( "\"all children are the same sex\" is basically saying \"all boys\" OR \"all girls\" (pick one side only)\r<BR>\n" );
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document.write( "There are 2 ways to either have all 4 boys or have all 4 girls. This is out of 16, so 2/16 = 1/8 = 0.125 = 12.5% is the answer for part b\r<BR>\n" );
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document.write( "c).there is at least one boy \r<BR>\n" );
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document.write( "There are 2 ways to compute this. We could follow\r<BR>\n" );
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document.write( "method 1: count the number of combinations with at least one B in them and we'd count out 15 of 16. So the probability is 15/16 = 0.9375 = 93.75%\r<BR>\n" );
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document.write( "OR\r<BR>\n" );
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document.write( "method 2: the probability of having all girls is 6.25% which is the same as the probability of having all boys. Subtract this from 100% to get 100% - 6.25% = 93.75%\r<BR>\n" );
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document.write( "No matter which method you ultimatley go with, the final answer is 93.75% which can be represented as the fraction 15/16\r<BR>\n" );
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document.write( "In summary, the answers are..\r<BR>\n" );
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document.write( "(a) 1/16 = 0.0625 = 6.25%<BR>\n" );
document.write( "(b) 1/8 = 0.125 = 12.5%<BR>\n" );
document.write( "(c) 15/16 = 0.9375 = 93.75%\r<BR>\n" );
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document.write( "I'm showing all 3 formats (fraction, decimal, percent) as the answer. Choose whichever answer format your teacher wants to use. </SPAN>\n" );
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