document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1084188>Question 1084188</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The probability that a person selected at random from a population will exhibit the classic symptom of a certain disease is 0.20 and the probability that a person selected at random has the disease is 0.23 the probability that a person who has the symptoms also has the disease is 0.18 A person is selected at random from a population does not have the symptoms.what is the probability that the person has the disease?? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #698257 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=698257\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"4\"><BR>\n" );
document.write( "Define the following events:<BR>\n" );
document.write( "S = person exhibits symptoms (they may or may not actually have the disease)<BR>\n" );
document.write( "D = person actually has the disease (they may or may not show symptoms)\r<BR>\n" );
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document.write( "Those are simple events which can be combined to form this compound event<BR>\n" );
document.write( "S and D = person has symptoms AND person actually has disease\r<BR>\n" );
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document.write( "We're given the following probabilities<BR>\n" );
document.write( "P(S) = 0.2<BR>\n" );
document.write( "P(D) = 0.23<BR>\n" );
document.write( "P(S and D) = P(D and S) = 0.18\r<BR>\n" );
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document.write( "We want to find the conditional probability P(D|S') which is another way of asking \"what is P(D) given that S' has occured?\". <BR>\n" );
document.write( "A translation: \"What is the probability of getting a person with the disease <u>given</u> they do not show symptoms?\"\r<BR>\n" );
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document.write( "Note: The notation <font color=blue>S'</font> is the complementary event of event S. If S doesn't happen, then S' does happen and vice versa. They are opposite one another. Since <BR>\n" );
document.write( "S = person exhibits symptoms<BR>\n" );
document.write( "this means<BR>\n" );
document.write( "S' = person does not show symptoms<BR>\n" );
document.write( "Your book may use the notation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Ec ALIGN=MIDDLE  ALT=\"S%5Ec\"   DISABLED_style= \"max-width:90%; height:auto;\" > instead of S' to denote a complementary event. As you can probably guess, the 'c' in the exponent means \"complementary\".\r<BR>\n" );
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document.write( "The conditional probability formula we'll use is<BR>\n" );
document.write( "P(D|S') = P(D and S')/P(S')\r<BR>\n" );
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document.write( "so we'll first need to find P(S') and P(D and S')\r<BR>\n" );
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document.write( "Let's find the probability of not showing symptoms P(S')<BR>\n" );
document.write( "P(S) + P(S') = 1<BR>\n" );
document.write( "P(S') = 1 - P(S)<BR>\n" );
document.write( "P(S') = 1 - 0.2<BR>\n" );
document.write( "P(S') = 0.8\r<BR>\n" );
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document.write( "Use the <a  rel=nofollow HREF=\"https://en.wikipedia.org/wiki/Law_of_total_probability\">Law of Total Probability</a> to find P(D and S')<BR>\n" );
document.write( "P(D) = P(D and S) + P(D and S')<BR>\n" );
document.write( "0.23 = 0.18 + P(D and S')<BR>\n" );
document.write( "P(D and S') = 0.05\r<BR>\n" );
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document.write( "Now we can compute the answer<BR>\n" );
document.write( "P(D|S') = P(D and S')/P(S')<BR>\n" );
document.write( "P(D|S') = 0.05/0.8<BR>\n" );
document.write( "P(D|S') = 0.0625\r<BR>\n" );
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document.write( "The probability is 0.0625\r<BR>\n" );
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document.write( "Notes:<BR>\n" );
document.write( "0.0625 = 625/10000 = 1/16 is the fraction form of the answer<BR>\n" );
document.write( "0.0625*(100%) = 6.25% is the percent form of the answer<BR>\n" );
document.write( "the percent, decimal and fraction forms all represent the same answer, just in different ways</font> </SPAN>\n" );
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