document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191943>Question 1191943</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the following probabilities to answer the question. Don't round; enter your answers as ratios and let WAMAP do the calculations for you. It may be helpful to make a contingency table.<BR>\n" );
document.write( "P(A)= 0.31, P(B)=0.76 and P( A and B)=0.18<BR>\n" );
document.write( "P(B | A)=<BR>\n" );
document.write( "p( Not B | A)=<BR>\n" );
document.write( "P( Not B | Not A)= </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823811 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=823811\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Given info:<BR>\n" );
document.write( "P(A) = 0.31<BR>\n" );
document.write( "P(B) = 0.76<BR>\n" );
document.write( "P(A and B) = 0.18\r<BR>\n" );
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document.write( "Based on the given info, we can say<BR>\n" );
document.write( "P(not A) = 1 - P(A) = 1 - 0.31 = 0.69<BR>\n" );
document.write( "P(not B) = 1 - P(B) = 1 - 0.76 = 0.24<BR>\n" );
document.write( "In short,<BR>\n" );
document.write( "P(not A) = 0.69<BR>\n" );
document.write( "P(not B) = 0.24\r<BR>\n" );
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document.write( "This is what the contingency table starts off looking like<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >A</TD><TD >not A</TD><TD >Total</TD></tr><tr><TD >B</TD><TD >P(A and B)</TD><TD >P(not A and B)</TD><TD >P(B)</TD></tr><tr><TD >not B</TD><TD >P(A and not B)</TD><TD >P(not A and not B)</TD><TD >P(not B)</TD></tr><tr><TD >Total</TD><TD >P(A)</TD><TD >P(not A)</TD><TD >1</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "Fill in the given values, and the values found so far<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >A</TD><TD >not A</TD><TD >Total</TD></tr><tr><TD >B</TD><TD >0.18</TD><TD >P(not A and B)</TD><TD >0.76</TD></tr><tr><TD >not B</TD><TD >P(A and not B)</TD><TD >P(not A and not B)</TD><TD >0.24</TD></tr><tr><TD >Total</TD><TD >0.31</TD><TD >0.69</TD><TD >1</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "In the \"A\" column, notice how we're missing one value in the middle.<BR>\n" );
document.write( "Apply the law of total probability<BR>\n" );
document.write( "P(A and B) + P(A and not B) = P(A)<BR>\n" );
document.write( "0.18 + P(A and not B) = 0.31<BR>\n" );
document.write( "P(A and not B) = 0.31 - 0.18<BR>\n" );
document.write( "P(A and not B) = 0.13\r<BR>\n" );
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document.write( "Then along the first row, we can say a similar argument.<BR>\n" );
document.write( "P(A and B) + P(not A and B) = P(B)<BR>\n" );
document.write( "0.18 + P(not A and B) = 0.76<BR>\n" );
document.write( "P(not A and B) = 0.76 - 0.18<BR>\n" );
document.write( "P(not A and B) = 0.58\r<BR>\n" );
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document.write( "Lastly, we can look at row 2 or column 2 to find that final missing probability.<BR>\n" );
document.write( "P(A and not B) + P(not A and not B) = P(not B)<BR>\n" );
document.write( "0.13 + P(not A and not B) = 0.24<BR>\n" );
document.write( "P(not A and not B) = 0.24 - 0.13<BR>\n" );
document.write( "P(not A and not B) = 0.11<BR>\n" );
document.write( "Or,<BR>\n" );
document.write( "P(not A and B) + P(not A and not B) = P(not A)<BR>\n" );
document.write( "0.58 + P(not A and not B) = 0.69<BR>\n" );
document.write( "P(not A and not B) = 0.69 - 0.58<BR>\n" );
document.write( "P(not A and not B) = 0.11\r<BR>\n" );
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document.write( "To summarize so far, we calculated:<BR>\n" );
document.write( "P(A and not B) = 0.13<BR>\n" );
document.write( "P(not A and B) = 0.58<BR>\n" );
document.write( "P(not A and not B) = 0.11\r<BR>\n" );
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document.write( "This is what the completed contigency table looks like <BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >A</TD><TD >not A</TD><TD >Total</TD></tr><tr><TD >B</TD><TD >0.18</TD><TD >0.58</TD><TD >0.76</TD></tr><tr><TD >not B</TD><TD >0.13</TD><TD >0.11</TD><TD >0.24</TD></tr><tr><TD >Total</TD><TD >0.31</TD><TD >0.69</TD><TD >1</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "Optionally if you wanted, you could convert those decimal values to their fractional forms.\r<BR>\n" );
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document.write( "Now to calculate the conditional probabilities\r<BR>\n" );
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document.write( "P(B | A) = P(B and A)/P(A)<BR>\n" );
document.write( "P(B | A) = P(A and B)/P(A)<BR>\n" );
document.write( "P(B | A) = 0.18/0.31<BR>\n" );
document.write( "P(B | A) = <font color=red>18/31</font>\r<BR>\n" );
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document.write( "P(not B | A) = P(not B and A)/P(A)<BR>\n" );
document.write( "P(not B | A) = P(A and not B)/P(A)<BR>\n" );
document.write( "P(not B | A) = 0.13/0.31<BR>\n" );
document.write( "P(not B | A) = <font color=red>13/31</font><BR>\n" );
document.write( "Or another route you could take is:<BR>\n" );
document.write( "P(not B | A) = 1 - P(B | A)<BR>\n" );
document.write( "P(not B | A) = 1 - 18/31<BR>\n" );
document.write( "P(not B | A) = 31/31 - 18/31<BR>\n" );
document.write( "P(not B | A) = (31-18)/31<BR>\n" );
document.write( "P(not B | A) = <font color=red>13/31</font><BR>\n" );
document.write( "This alternative route works because P(B | A) and P(not B|A) are complementary events in the same way that P(B) and P(not B) are complementary.\r<BR>\n" );
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document.write( "P(not B | not A) = P(not B and not A)/P(not A)<BR>\n" );
document.write( "P(not B | not A) = P(not A and not B)/P(not A)<BR>\n" );
document.write( "P(not B | not A) = 0.11/0.69<BR>\n" );
document.write( "P(not B | not A) = <font color=red>11/69</font>\r<BR>\n" );
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document.write( "---------------------------------------------------------\r<BR>\n" );
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document.write( "Answers:\r<BR>\n" );
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document.write( "P(B | A) = <font color=red>18/31</font><BR>\n" );
document.write( "P(not B | A) = <font color=red>13/31</font><BR>\n" );
document.write( "P(not B | not A) = <font color=red>11/69</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );