document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=498471>Question 498471</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factories A and B produce computers. Factory A produces 2 times as many computers as factory B. The probability that an item produced by factory A is defective is 0.021 and the probability that an item produced by factory B is defective is 0.024.<BR>\n" );
document.write( "A computer is selected at random and it is found to be defective. What is the probability it came from factory A? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #337050 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=337050\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x be the total number of computers that are built.<BR>\n" );
document.write( "since 2 out of 3 come from factory A, then factory A builds (2/3)*x and factory B builds (1/3)*x.<BR>\n" );
document.write( "the number of defective computers that come from factory A are .021 * (2/3) * x.<BR>\n" );
document.write( "the number of defective computers that come from factory B are .024 * (1/3) * x.<BR>\n" );
document.write( "simplifying these expressions gets:<BR>\n" );
document.write( "the number of defective computers that come from factory A are .014 * x.<BR>\n" );
document.write( "the number of defective computers that come from factory B are .008 * x.<BR>\n" );
document.write( "therefore:<BR>\n" );
document.write( "the probability that a defective computer comes from factory A is .014.<BR>\n" );
document.write( "the probability that a defective computer comes from factory B is .008<BR>\n" );
document.write( "the probability of getting a defective computer is .014 + .008 = .022.\r<BR>\n" );
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document.write( "as an example:<BR>\n" );
document.write( "assume that 3000 computers were built.<BR>\n" );
document.write( "2000 came from factory A.<BR>\n" );
document.write( "1000 came from factory B.<BR>\n" );
document.write( ".021 * 2000 = 42 were defective from factory A.<BR>\n" );
document.write( ".024 * 1000 = 24 were defective from factory B.<BR>\n" );
document.write( "total defective were 66.<BR>\n" );
document.write( "probability of getting a defective computer from factory A is 42/3000 = .014<BR>\n" );
document.write( "probability of getting a defective computer from factory B is 24/3000 = .008<BR>\n" );
document.write( "probability of getting a defective computer is 66/3000 = .022\r<BR>\n" );
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document.write( "the question to be answered is:<BR>\n" );
document.write( "A computer is selected at random and it is found to be defective. What is the probability it came from factory A?<BR>\n" );
document.write( "This can be translated to:<BR>\n" );
document.write( "given that a computer is defective, what is the probability that it came from factory A?<BR>\n" );
document.write( "the formula for p(X given Y) is equal to p(X intersect Y) / p(Y)<BR>\n" );
document.write( "in this problem we have:<BR>\n" );
document.write( "p(X) = probability that a defective computer came from factory A.<BR>\n" );
document.write( "p(Y) = probability that a defective computer came from factory A or factory B.<BR>\n" );
document.write( "p(X intersect Y) = probability that a defective computer came from factory A and also came from factory A or factory B.  That's the same as the probability that it came from factory A.<BR>\n" );
document.write( "the formula of p(X intersect Y) / p(Y) beecomes .014/.022 = .6363636363 = .64\r<BR>\n" );
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document.write( "in our example we had:<BR>\n" );
document.write( "3000 computers were built.<BR>\n" );
document.write( "2000 came from factory A<BR>\n" );
document.write( "1000 came from factory B<BR>\n" );
document.write( "42 were defective from factory A<BR>\n" );
document.write( "24 were defective from factory B<BR>\n" );
document.write( "66 were defective from factory A or B\r<BR>\n" );
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document.write( "given that the computer was defective, then it had to be one of the 66 defective computers in the lot.<BR>\n" );
document.write( "since there are 42 out of 66 defective that are from factory A, then the probability that a defective computer came from factory A would be 42/66 = .636363636 = .64 rounded to the nearest hundredth.\r<BR>\n" );
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document.write( "i believe this is the answer you are looking for.<BR>\n" );
document.write( "let me know if otherwise.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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