document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205234>Question 1205234</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a recent year 8,073,000 male students and 10,980,000 female students were<BR>\n" );
document.write( "enrolled as undergraduates. Receiving aid were 60.6% of the male students and<BR>\n" );
document.write( "65.2% of the female students. Of those receiving aid, 44.8% of the males got<BR>\n" );
document.write( "federal aid and 50.4% of the females got federal aid. Choose 1 student at random.<BR>\n" );
document.write( "(Hint: Make a tree diagram.) Find the probability that the student is<BR>\n" );
document.write( "a. A male student without aid<BR>\n" );
document.write( "b. A male student, given that the student has aid<BR>\n" );
document.write( "c. A female student or a student who receives federal aid </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #842319 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=842319\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll use a two-way table instead of a tree diagram.\r<BR>\n" );
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document.write( "Here's the starting template<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >Male</TD><TD >Female</TD><TD >Total</TD></tr><tr><TD >Aid</TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >No Aid</TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >Total</TD><TD ></TD><TD ></TD><TD ></TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "Place the values 8,073,000 and 10,980,000 at the bottom row of the male and female columns respectively.\r<BR>\n" );
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document.write( "60.6% of 8,073,000 = 0.606*(8,073,000) = 4,892,238 male students got aid.<BR>\n" );
document.write( "65.2% of 10,980,000 = 0.652*(10,980,000) = 7,158,960 female students got aid.\r<BR>\n" );
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document.write( "Let's place those results into the table.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >Male</TD><TD >Female</TD><TD >Total</TD></tr><tr><TD >Aid</TD><TD >4,892,238</TD><TD >7,158,960</TD><TD >12,051,198</TD></tr><tr><TD >No Aid</TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >Total</TD><TD >8,073,000</TD><TD >10,980,000</TD><TD >19,053,000</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "To finish up the table, subtract the count who got aid from the total.<BR>\n" );
document.write( "Eg: 8,073,000 - 4,892,238 = 3,180,762 male students did not get aid.\r<BR>\n" );
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document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >Male</TD><TD >Female</TD><TD >Total</TD></tr><tr><TD >Aid</TD><TD >4,892,238</TD><TD >7,158,960</TD><TD >12,051,198</TD></tr><tr><TD >No Aid</TD><TD >3,180,762</TD><TD >3,821,040</TD><TD >7,001,802</TD></tr><tr><TD >Total</TD><TD >8,073,000</TD><TD >10,980,000</TD><TD >19,053,000</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "Of those who got aid, we have this sub-table<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >Male</TD><TD >Female</TD><TD >Total</TD></tr><tr><TD >Federal</TD><TD >2,191,723</TD><TD >3,608,116</TD><TD >5,799,839</TD></tr><tr><TD >Not federal</TD><TD >2,700,515</TD><TD >3,550,844</TD><TD >6,251,359</TD></tr><tr><TD >Total</TD><TD >4,892,238</TD><TD >7,158,960</TD><TD >12,051,198</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "Example: 44.8% of 4,892,238 = 0.448*4892238 = 2191722.624 which rounds to 2191723\r<BR>\n" );
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document.write( "The values in the bottom row match up with the values in row 1 of the previous table. Spreadsheet software is strongly recommended.\r<BR>\n" );
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document.write( "-------------------------------------------------\r<BR>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "Refer to the first table<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >Male</TD><TD >Female</TD><TD >Total</TD></tr><tr><TD >Aid</TD><TD >4,892,238</TD><TD >7,158,960</TD><TD >12,051,198</TD></tr><tr><TD >No Aid</TD><TD >3,180,762</TD><TD >3,821,040</TD><TD >7,001,802</TD></tr><tr><TD >Total</TD><TD >8,073,000</TD><TD >10,980,000</TD><TD >19,053,000</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "There are 3,180,762 men who did not get aid out of 19,053,000 students.\r<BR>\n" );
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document.write( "3180762/19053000 = <font color=red>0.166943</font> approximately\r<BR>\n" );
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document.write( "-------------------------------------------------\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Refer to the same table as mentioned in part (a)\r<BR>\n" );
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document.write( "The key term \"given\" means we know 100% that whoever is randomly selected, they received aid of some kind (federal or non-federal).<BR>\n" );
document.write( "We focus entirely on the \"aid\" row.\r<BR>\n" );
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document.write( "There are 4,892,238 men who got aid out of 12,051,198 students who got aid.\r<BR>\n" );
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document.write( "4892238/12051198 = <font color=red>0.405954</font> approximately\r<BR>\n" );
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document.write( "-------------------------------------------------\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "We'll need to use this table<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >Male</TD><TD >Female</TD><TD >Total</TD></tr><tr><TD >Aid</TD><TD >4,892,238</TD><TD >7,158,960</TD><TD >12,051,198</TD></tr><tr><TD >No Aid</TD><TD >3,180,762</TD><TD >3,821,040</TD><TD >7,001,802</TD></tr><tr><TD >Total</TD><TD >8,073,000</TD><TD >10,980,000</TD><TD >19,053,000</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "and this table as well<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >Male</TD><TD >Female</TD><TD >Total</TD></tr><tr><TD >Federal</TD><TD >2,191,723</TD><TD >3,608,116</TD><TD >5,799,839</TD></tr><tr><TD >Not federal</TD><TD >2,700,515</TD><TD >3,550,844</TD><TD >6,251,359</TD></tr><tr><TD >Total</TD><TD >4,892,238</TD><TD >7,158,960</TD><TD >12,051,198</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "There's probably a clever way to combine the two tables, but my mind is drawing a blank right now.\r<BR>\n" );
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document.write( "There are 10,980,000 female students and 5,799,839 students who got federal aid. The overlap is the 3,608,116 female students who got federal aid.\r<BR>\n" );
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document.write( "Use the inclusion exclusion principle to say that there are <BR>\n" );
document.write( "10980000+5799839-3608116 = 13171723<BR>\n" );
document.write( "students who are female, got federal aid, or both.\r<BR>\n" );
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document.write( "This is out of the 19053000 students total.<BR>\n" );
document.write( "13171723/19053000 = <font color=red>0.69132</font> approximately.\r<BR>\n" );
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document.write( "-----------------------------------------------------------<BR>\n" );
document.write( "-----------------------------------------------------------\r<BR>\n" );
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document.write( "Summary<BR>\n" );
document.write( "(a) <font color=red>0.166943</font><BR>\n" );
document.write( "(b) <font color=red>0.405954</font> <BR>\n" );
document.write( "(c) <font color=red>0.69132</font><BR>\n" );
document.write( "Each value is approximate. Round them however needed.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );