document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172449>Question 1172449</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the contingency table below to find the following probabilities.<BR>\n" );
document.write( "a.​ A|B<BR>\n" );
document.write( "b.​ A|B'<BR>\n" );
document.write( "c.​ A'|B'<BR>\n" );
document.write( "d. Are events A and B​ independent?\r<BR>\n" );
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document.write( " Table_Data,B,B`<BR>\n" );
document.write( "A,30,40<BR>\n" );
document.write( "A',40,50<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 #797491 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=797491\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Original Table<BR>\n" );
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document.write( "<tr><TD ></TD><TD >B</TD><TD >B'</TD></tr><tr><TD >A</TD><TD >30</TD><TD >40</TD></tr><tr><TD >A'</TD><TD >40</TD><TD >50</TD></tr>\n" );
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document.write( "Compute the subtotals and the grand total<BR>\n" );
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document.write( "<tr><TD ></TD><TD >B</TD><TD >B'</TD><TD >Total</TD></tr><tr><TD >A</TD><TD >30</TD><TD >40</TD><TD >70</TD></tr><tr><TD >A'</TD><TD >40</TD><TD >50</TD><TD >90</TD></tr><tr><TD >Total</TD><TD >70</TD><TD >90</TD><TD >160</TD></tr>\n" );
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document.write( "Divide each item by the grand total 160, and fully reduce, to compute the probabilities<BR>\n" );
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document.write( "<tr><TD ></TD><TD >B</TD><TD >B'</TD><TD >Total</TD></tr><tr><TD >A</TD><TD >3/16</TD><TD >1/4</TD><TD >7/16</TD></tr><tr><TD >A'</TD><TD >1/4</TD><TD >5/16</TD><TD >9/16</TD></tr><tr><TD >Total</TD><TD >7/16</TD><TD >9/16</TD><TD >1</TD></tr>\n" );
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document.write( "We'll use this probability table to answer parts (a) through (d)\r<BR>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "P(A and B) = 3/16 .... upper left corner of table<BR>\n" );
document.write( "P(B) = 7/16 .... bottom of column 1\r<BR>\n" );
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document.write( "P(A | B) = Probability of A, given B<BR>\n" );
document.write( "P(A | B) = P(A and B)/P(B)<BR>\n" );
document.write( "P(A | B) = (3/16)/(7/16)<BR>\n" );
document.write( "P(A | B) = (3/16)*(16/7)<BR>\n" );
document.write( "P(A | B) = 3/7\r<BR>\n" );
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document.write( "<font color=red>Answer: 3/7</font><BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "P(A and B') = 1/4<BR>\n" );
document.write( "P(B') = 9/16\r<BR>\n" );
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document.write( "P(A given B') = P(A and B')/P(B')<BR>\n" );
document.write( "P(A given B') = (1/4) divide by (9/16)<BR>\n" );
document.write( "P(A given B') = (1/4)*(16/9)<BR>\n" );
document.write( "P(A given B') = (1*16)/(4*9)<BR>\n" );
document.write( "P(A given B') = 16/36<BR>\n" );
document.write( "P(A given B') = (4*4)/(4*9)<BR>\n" );
document.write( "P(A given B') = 4/9\r<BR>\n" );
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document.write( "<font color=red>Answer: 4/9</font>\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "P(A' and B') = 5/16<BR>\n" );
document.write( "P(B') = 9/16\r<BR>\n" );
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document.write( "P(A' given B') = P(A' and B')/P(B')<BR>\n" );
document.write( "P(A' given B') = (5/16) divide by (9/16)<BR>\n" );
document.write( "P(A' given B') = (5/16)*(16/9)<BR>\n" );
document.write( "P(A' given B') = 5/9\r<BR>\n" );
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document.write( "<font color=red>Answer: 5/9</font><BR>\n" );
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document.write( "Part (d)\r<BR>\n" );
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document.write( "Events A and B would be independent if and only if the following two items are true<BR>\n" );
document.write( "P(A given B) = P(A)<BR>\n" );
document.write( "P(B given A) = P(B)<BR>\n" );
document.write( "Independent events are not linked together. If A and B are independent, then one event occurring does not change the probability of the other.\r<BR>\n" );
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document.write( "From part (a), we found P(A given B) = 3/7, but this is not the same value as P(A) = 7/16, which is what the table shows. This concludes that A and B are not independent.\r<BR>\n" );
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document.write( "As an alternative, we could also use the equation<BR>\n" );
document.write( "P(A and B) = P(A)*P(B)<BR>\n" );
document.write( "to find that<BR>\n" );
document.write( "P(A and B) = P(A)*P(B)<BR>\n" );
document.write( "3/16 = (7/16)*(7/16)<BR>\n" );
document.write( "3/16 = 49/256<BR>\n" );
document.write( "which is a false equation, so events A and B are not independent.\r<BR>\n" );
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document.write( "<font color=red>Answer: A and B are <u>not</u> independent. They are dependent.</font></font> </SPAN>\n" );
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