document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1160215>Question 1160215</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Urgent Help Needed\r<BR>\n" );
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document.write( "Below is given the summary from the 112th Congress of Senators whose terms end in 2013, 2015, or 2017.\r<BR>\n" );
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document.write( "	2013	2015	2017<BR>\n" );
document.write( "Democrat	21	20	1<BR>\n" );
document.write( "Republican	8	15	13\r<BR>\n" );
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document.write( "Choose one of these Senators at random and find<BR>\n" );
document.write( "a)	P(Democrat and term expires in 2015)<BR>\n" );
document.write( "b)	P(Republican or term expires in 2013)<BR>\n" );
document.write( "c)	P(Republican given term expires in 2017)<BR>\n" );
document.write( "d)	Are the events “Republican” and “term expires in 2015” independent? Explain.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #783469 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=783469\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the following concepts apply.\r<BR>\n" );
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document.write( "p(a or b) = p(a) + p(b) - p(a and b).\r<BR>\n" );
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document.write( "p(a and b) = p(a) * p(b) if the events are independent.\r<BR>\n" );
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document.write( "p(a given b) = p(a and b) / p(b).\r<BR>\n" );
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document.write( "p(a given b) = p(a0 if the events are independent.\r<BR>\n" );
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document.write( "if you draw a table of what you are given, you would find the table looks like this:\r<BR>\n" );
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document.write( "politics             2013          2015         2017         total\r\n" );
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document.write( "democrat             21            21           1             42\r\n" );
document.write( "republican           8             15           13            36\r\n" );
document.write( "total                29            36           14            78\r\n" );
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document.write( "let d be the event that the person is democrat.<BR>\n" );
document.write( "let r be the event that the person is republican.<BR>\n" );
document.write( "let 2013 be the event that the person retires in 2013.<BR>\n" );
document.write( "let 2015 be the event that the person retires in 2015.<BR>\n" );
document.write( "let 2017 be the event that the person retires in 2017.\r<BR>\n" );
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document.write( "from the table:<BR>\n" );
document.write( "p(d) = 42 / 78 because there are 42 democrats out of a total of 78 senators who retired in either 2013 or 2015 or 2017.<BR>\n" );
document.write( "p(r) = 36 / 78 because there are 36 republicans out of a total of 78 senators who retired in either 2013 or 2015 or 2017.\r<BR>\n" );
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document.write( "p(2013) = 29 because there are 29 senators who retired in 2013 out of a total of 78 senators who retired in all 3 years<BR>\n" );
document.write( "p(2015) = 35 because there are 35 senators who retired in 2015 out of a total of 78 senators who retired in all 3 years.<BR>\n" );
document.write( "p(2017) = 14 because there are 14 senators who retired in 2017 out of a total of 78 senators who retired in all 3 years.\r<BR>\n" );
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document.write( "your universe is 78 senators who retired in all 3 years of (2013, 2015, 2017).<BR>\n" );
document.write( "42 are democrats and 36 are republicans.<BR>\n" );
document.write( "29 retired in 2013 and 35 retired in 2015 and 14 retired in 2017.\r<BR>\n" );
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document.write( "your questions are:\r<BR>\n" );
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document.write( "a) P(Democrat and term expires in 2015)<BR>\n" );
document.write( "b) P(Republican or term expires in 2013)<BR>\n" );
document.write( "c) P(Republican given term expires in 2017)<BR>\n" );
document.write( "d) Are the events “Republican” and “term expires in 2015” independent? Explain.\r<BR>\n" );
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document.write( "answer to a):<BR>\n" );
document.write( "from the table, you can see that there are 20 democrats who retired in 2015.<BR>\n" );
document.write( "therefore p(d and 2015) = 20/78.\r<BR>\n" );
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document.write( "answer to b):<BR>\n" );
document.write( "this one is asking for p(r or 2013)<BR>\n" );
document.write( "the formula to use here is p(r or 2013) = p(r) + p(2013) - p(r and 2013).<BR>\n" );
document.write( "from the table, you can see that:<BR>\n" );
document.write( "p(r) = 36/78<BR>\n" );
document.write( "p(2013) = 29/78<BR>\n" );
document.write( "p(r and 2013) = 8<BR>\n" );
document.write( "therefore:<BR>\n" );
document.write( "p(r or 2013) is equal to p(r) + p(2013) - p(r and 2013) which is equal to 36/78 + 29/78 - 8/78 = 57/78.\r<BR>\n" );
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document.write( "answer to c):<BR>\n" );
document.write( "this is one is asking for p(r given 2017)<BR>\n" );
document.write( "the formula to use here is p(r given 2017) = p(r and 2017) / p(2017)<BR>\n" );
document.write( "from the table, you can see that:<BR>\n" );
document.write( "p(2017) = 14/78<BR>\n" );
document.write( "p(r and 2017) = 13/78<BR>\n" );
document.write( "therefore:<BR>\n" );
document.write( "p(r given 2017) = p(r and 2017) / p(2017) = (13/78) / (14/78) = 13/14.\r<BR>\n" );
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document.write( "answer to d):<BR>\n" );
document.write( "it they are independent, then p(r given 2015) would be equal to p(r).<BR>\n" );
document.write( "why is this so?<BR>\n" );
document.write( "if they were independent, then p(r and 2017) would be equal to p(r) * p(2017).<BR>\n" );
document.write( "therefore p(r given 2017) would be equal to p(r and 2017) / p(2017) which would be equal to p(r) * p(2017) / p(2017) which would be equal to p(r).<BR>\n" );
document.write( "from the table, you can see that p(r) = 36/78.<BR>\n" );
document.write( "from the table, you can see that p(r and 2017) = 13/78) and p(2017) = 14/78.<BR>\n" );
document.write( "therefore p(r given 2017) = p(r and 2017) / p(2017) = (13/78) / (14/78) which is equal to 13/14 which is not equal to p(r) = 36/78<BR>\n" );
document.write( "therefore the two events are not independent.\r<BR>\n" );
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