document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178009>Question 1178009</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a certain population the probability that a randomly selected subject will have been exposed<BR>\n" );
document.write( "to a certain allergen and experience a reaction to the allergen is .60. The probability is .8 that a<BR>\n" );
document.write( "subject exposed to the allergen will experience an allergic reaction. If a subject is selected at random from this population, what is the probability that he or she will have been exposed to the<BR>\n" );
document.write( "allergen? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850388 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=850388\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem using probability notation.\r<BR>\n" );
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document.write( "**Define Events**\r<BR>\n" );
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document.write( "* **E:** Event that a subject has been exposed to the allergen.<BR>\n" );
document.write( "* **R:** Event that a subject experiences an allergic reaction.\r<BR>\n" );
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document.write( "**Given Information**\r<BR>\n" );
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document.write( "* P(E ∩ R) = 0.60 (Probability of being exposed AND having a reaction)<BR>\n" );
document.write( "* P(R | E) = 0.80 (Probability of having a reaction GIVEN exposure)\r<BR>\n" );
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document.write( "**Goal**\r<BR>\n" );
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document.write( "* Find P(E) (Probability of being exposed to the allergen)\r<BR>\n" );
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document.write( "**Apply Conditional Probability Formula**\r<BR>\n" );
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document.write( "We know that:\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( "We can rearrange this formula to solve for P(E):\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( "**Substitute the Given Values**\r<BR>\n" );
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document.write( "* P(E) = 0.60 / 0.80<BR>\n" );
document.write( "* P(E) = 6/8<BR>\n" );
document.write( "* P(E) = 3/4<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.**<BR>\n" );
document.write( " </SPAN>\n" );
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