document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=413498>Question 413498</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose D, E, and F are independent events with P(D) = 0.21 , P(E) = 0.48, and P(F) = 0.18. What is the probability that at least one of the three events occurs? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #290160 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=290160\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The probability that at least 1 of the events occur is equal to 1 minus the probability that none of the events occur.\r<BR>\n" );
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document.write( "p of d = .21<BR>\n" );
document.write( "p of e = .48<BR>\n" );
document.write( "p of f = .18\r<BR>\n" );
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document.write( "p of not d = 1 - .21 = .79<BR>\n" );
document.write( "p of not e = 1 - .48 = .52<BR>\n" );
document.write( "p of not f = 1 - .18 = .82\r<BR>\n" );
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document.write( "p of none of the events occurring is equal to .79 * .52 * .82 = .336856\r<BR>\n" );
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document.write( "p of at least one of the event occurring would be 1 -.336856 = .663144\r<BR>\n" );
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document.write( "to see if this is good, just take the possibility of 1, 2, or 3 of the events occurring and add them up.\r<BR>\n" );
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document.write( "it's a lot more labor intensive to do it this way, but it is instructive.\r<BR>\n" );
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document.write( "because each of the probabilities are different, you can't take any shortcuts.\r<BR>\n" );
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document.write( "you have to find:\r<BR>\n" );
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document.write( "p of only first event occurring.<BR>\n" );
document.write( "p of only second event occurring.<BR>\n" );
document.write( "p of only third event occurring.\r<BR>\n" );
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document.write( "p of only first and second event occurring.<BR>\n" );
document.write( "p of only first and third event occurring.<BR>\n" );
document.write( "p of only second and third event occurring.\r<BR>\n" );
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document.write( "p of all 3 events occurring.\r<BR>\n" );
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document.write( "then you have to add them up.\r<BR>\n" );
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document.write( "the number required are shown below:\r<BR>\n" );
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document.write( ".089544 = p of only first event occurring.<BR>\n" );
document.write( ".310944 = p of only second event occurring.<BR>\n" );
document.write( ".073944 = p of only third event occurring.\r<BR>\n" );
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document.write( ".082656 = p of only first and second event occurring.<BR>\n" );
document.write( ".019656 = p of only first and third event occurring.<BR>\n" );
document.write( ".068256 = p of only second and third event occurring.\r<BR>\n" );
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document.write( ".018144 p of all 3 events occurring.\r<BR>\n" );
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document.write( "sum of all these probabilities is equal to .663144\r<BR>\n" );
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document.write( "That's the same we got the easy way by taking 1 - p that none of the events occur.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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