document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1088263>Question 1088263</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a person is vaccinated properly, the probability of his getting a certain disease is 0.05.  Without a vaccination, the probability of getting the disease is 0.35.  Assume that 1/3 of the population is vaccinated.  If a person is selected at random from the population, what is the probability of that person's getting 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 #702511 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=702511\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">V = person is vaccinated<BR>\n" );
document.write( "D = person gets disease\r<BR>\n" );
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document.write( "Given Info: \"If a person is vaccinated properly, the probability of his getting a certain disease is 0.05\"\r<BR>\n" );
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document.write( "Translation to symbols: P(D|V) = 0.05\r<BR>\n" );
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document.write( "This is conditional probability notation. Asking \"what is P(D|V)?\" is the same as asking \"What is P(D) if we know V is true?\"<BR>\n" );
document.write( "In English, it means \"What is the probability of getting the disease <u>given</u> the person is vaccinated\". The keyword \"given\" is important.\r<BR>\n" );
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document.write( "Given Info: \"Without a vaccination, the probability of getting the disease is 0.35\"\r<BR>\n" );
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document.write( "Translation to symbols: P(D|V') = 0.35\r<BR>\n" );
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document.write( "This is the same as \"the probability of getting the disease <u>given</u> the person is not vaccinated\".\r<BR>\n" );
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document.write( "V' is the complement of event V, which means P(V)+P(V') = 1 and P(V') = 1-P(V)\r<BR>\n" );
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document.write( "Given Info: \"Assume that 1/3 of the population is vaccinated\"\r<BR>\n" );
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document.write( "Translation: P(V) = 1/3\r<BR>\n" );
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document.write( "This means that P(V') = 1-P(V) = 1-(1/3) = 2/3\r<BR>\n" );
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document.write( "Summary so far:\r<BR>\n" );
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document.write( "P(D|V) = 0.05<BR>\n" );
document.write( "P(D|V') = 0.35<BR>\n" );
document.write( "P(V) = 1/3<BR>\n" );
document.write( "P(V') = 1/3\r<BR>\n" );
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document.write( "Given Question: \"If a person is selected at random from the population, what is the probability of that person's getting the disease?\"\r<BR>\n" );
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document.write( "What we want to find: The value of P(D)\r<BR>\n" );
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document.write( "Before we can find P(D), we need to do a bit of side work. \r<BR>\n" );
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document.write( "Let's use the definition of conditional probability to rewrite things a bit\r<BR>\n" );
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document.write( "P(D|V) = [P(D and V)]/P(V)<BR>\n" );
document.write( "P(D|V)*P(V) = P(D and V)<BR>\n" );
document.write( "P(D and V) = P(D|V)*P(V) ... <font color=blue>Call this equation (1)</font>\r<BR>\n" );
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document.write( "P(D|V') = [P(D and V')]/P(V')<BR>\n" );
document.write( "P(D|V')*P(V') = P(D and V')<BR>\n" );
document.write( "P(D and V') = P(D|V')*P(V') ... <font color=green>Call this equation (2)</font>\r<BR>\n" );
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document.write( "We'll use <font color=blue>equation (1)</font> and <font color=green>equation (2)</font> in the next section.\r<BR>\n" );
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document.write( "Now we can find P(D).\r<BR>\n" );
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document.write( "Using the <a  rel=nofollow HREF=\"https://en.wikipedia.org/wiki/Law_of_total_probability\">law of total probability</a>, we can say,\r<BR>\n" );
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document.write( "P(D) = P(D and V) + P(D and V')<BR>\n" );
document.write( "P(D) = <font color=blue>P(D|V)*P(V)</font> + P(D and V') ... Make a substitution using <font color=blue>equation (1)</font><BR>\n" );
document.write( "P(D) = P(D|V)*P(V) + <font color=green>P(D|V')*P(V')</font> ... Make a substitution using <font color=green>equation (2)</font><BR>\n" );
document.write( "P(D) = 0.05*(1/3) + 0.35*(2/3) ... plug in the given information (see the \"summary so far\" section above)<BR>\n" );
document.write( "P(D) = 0.25\r<BR>\n" );
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document.write( "In decimal form, the final answer is 0.25\r<BR>\n" );
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document.write( "In fraction form, the final answer is 1/4 (because 1/4 = 0.25)\r<BR>\n" );
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document.write( "In percent form, the final answer is 25%\r<BR>\n" );
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