document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172434>Question 1172434</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the contingency table to the right to determine the probability of events. <BR>\n" );
document.write( "a. What is the probability of event​ A?<BR>\n" );
document.write( "b. What is the probability of event ​A'?<BR>\n" );
document.write( "c. What is the probability of event A and​ B?<BR>\n" );
document.write( "d. What is the probability of event A or​ B?\r<BR>\n" );
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document.write( "Table_Data	B	B`<BR>\n" );
document.write( "A	        80      80<BR>\n" );
document.write( "A'       	50	70 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #797457 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=797457\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3>Part (a)\r<BR>\n" );
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document.write( "Given table<BR>\n" );
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document.write( "<tr><TD ></TD><TD >B</TD><TD >B’</TD></tr><tr><TD >A</TD><TD >80</TD><TD >80</TD></tr><tr><TD >A’</TD><TD >50</TD><TD >70</TD></tr>\n" );
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document.write( "Compute the totals for each row and column. Then find the grand total to be 280, which is in the bottom right corner. <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 >80</TD><TD >80</TD><TD >160</TD></tr><tr><TD >A’</TD><TD >50</TD><TD >70</TD><TD >120</TD></tr><tr><TD >Total</TD><TD >130</TD><TD >150</TD><TD >280</TD></tr>\n" );
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document.write( "If we divide everything by the grand total 280, and reduce fully, we get this table of 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 >2/7</TD><TD >2/7</TD><TD >4/7</TD></tr><tr><TD >A’</TD><TD >5/28</TD><TD >1/4</TD><TD >3/7</TD></tr><tr><TD >Total</TD><TD >13/28</TD><TD >15/28</TD><TD >1</TD></tr>\n" );
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document.write( "From the first row, we see<BR>\n" );
document.write( "P(A and B) = 2/7<BR>\n" );
document.write( "P(A and B') = 2/7\r<BR>\n" );
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document.write( "Which means:<BR>\n" );
document.write( "P(A) = P(A and B) + P(A and B')<BR>\n" );
document.write( "P(A) = 2/7 + 2/7<BR>\n" );
document.write( "P(A) = 4/7\r<BR>\n" );
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document.write( "This matches with the total shown in the table for row 1.\r<BR>\n" );
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document.write( "<font color=red>Answer: 4/7</font>\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Apply the law of probability complements<BR>\n" );
document.write( "P(A) + P(A') = 1<BR>\n" );
document.write( "P(A') = 1 - P(A)<BR>\n" );
document.write( "P(A') = 1 - (4/7)<BR>\n" );
document.write( "P(A') = (7/7) - (4/7)<BR>\n" );
document.write( "P(A') = (7-4)/7<BR>\n" );
document.write( "P(A') = 3/7<BR>\n" );
document.write( "This matches with the total for the A' row.\r<BR>\n" );
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document.write( "As an alternative approach:<BR>\n" );
document.write( "P(A') = P(A' and B) + P(A' and B')<BR>\n" );
document.write( "P(A') = 5/28 + 1/4<BR>\n" );
document.write( "P(A') = 5/28 + 7/28<BR>\n" );
document.write( "P(A') = 12/28<BR>\n" );
document.write( "P(A') = 3/7\r<BR>\n" );
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document.write( "<font color=red>Answer: 3/7</font>\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "Refer to the probability table in part (a). Look in the upper left corner which corresponds to the A row and B column.\r<BR>\n" );
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document.write( "<font color=red>Answer: 2/7</font>\r<BR>\n" );
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document.write( "Part (d)\r<BR>\n" );
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document.write( "P(B) = P(B and A) + P(B and A')<BR>\n" );
document.write( "P(B) = 2/7+5/28<BR>\n" );
document.write( "P(B) = 8/28+5/28<BR>\n" );
document.write( "P(B) = 13/28<BR>\n" );
document.write( "This matches with the total shown at the bottom of column B in the probability table back in part (a).\r<BR>\n" );
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document.write( "P(A or B) = P(A) + P(B) - P(A and B)<BR>\n" );
document.write( "P(A or B) = 4/7 + 13/28 - 2/7<BR>\n" );
document.write( "P(A or B) = 16/28 + 13/28 - 8/28<BR>\n" );
document.write( "P(A or B) = (16+13-8)/28<BR>\n" );
document.write( "P(A or B) = 21/28<BR>\n" );
document.write( "P(A or B) = 3/4\r<BR>\n" );
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document.write( "<font color=red>Answer: 3/4</font></font> </SPAN>\n" );
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