document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186892>Question 1186892</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a certain population the probability that a randomly selected subject will have been <BR>\n" );
document.write( "exposed to a certain allergen and experience a reaction to the allergen is .60. The <BR>\n" );
document.write( "probability is .8 that a subject exposed to the allergen will experience an allergic <BR>\n" );
document.write( "reaction. If a subject is selected at random from this population, what is the probability <BR>\n" );
document.write( "that he or she will have been exposed to the allergen? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849335 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849335\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem using conditional probability:\r<BR>\n" );
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document.write( "Let's define the events:\r<BR>\n" );
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document.write( "*   E: The subject has been exposed to the allergen.<BR>\n" );
document.write( "*   R: The subject experiences a reaction to the allergen.\r<BR>\n" );
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document.write( "We are given the following probabilities:\r<BR>\n" );
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document.write( "*   P(E ∩ R) = 0.60 (Probability of exposure *and* reaction)<BR>\n" );
document.write( "*   P(R|E) = 0.80 (Probability of reaction *given* exposure)\r<BR>\n" );
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document.write( "We want to find P(E), the probability that a subject has been exposed to the allergen.\r<BR>\n" );
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document.write( "We can use the formula for conditional probability:\r<BR>\n" );
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document.write( "P(R|E) = P(E ∩ R) / P(E)\r<BR>\n" );
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document.write( "Rearranging this formula to solve for P(E), we get:\r<BR>\n" );
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document.write( "P(E) = P(E ∩ R) / P(R|E)\r<BR>\n" );
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document.write( "Now, plug in the given values:\r<BR>\n" );
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document.write( "P(E) = 0.60 / 0.80<BR>\n" );
document.write( "P(E) = 0.75\r<BR>\n" );
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document.write( "Therefore, the probability that a randomly selected subject will have been exposed to the allergen is 0.75, or 75%.<BR>\n" );
document.write( " </SPAN>\n" );
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