document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203912>Question 1203912</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two machines turn out all the products in a factory, with the first machine producing 57% of the product and the second machine producing the rest. The first machine produces defective products 6% of the time and the second machine 2% of the time. What is the probability that a part made in this factory is not defective? (Enter answer as a decimal with at least 4 correct decimal places)<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839817 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=839817\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "A = 1st machine is selected<BR>\n" );
document.write( "B = 2nd machine is selected<BR>\n" );
document.write( "P(A) = 0.57<BR>\n" );
document.write( "P(B) = 1-P(A) = 1-0.57 = 0.43\r<BR>\n" );
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document.write( "D = part is defective<BR>\n" );
document.write( "~D = part is not defective<BR>\n" );
document.write( "P(D given A) = 0.06<BR>\n" );
document.write( "P(~D given A) = 1-P(D given A) = 1-0.06 = 0.94<BR>\n" );
document.write( "and,<BR>\n" );
document.write( "P(D given B) = 0.02<BR>\n" );
document.write( "P(~D given B) = 1-P(D given B) = 1-0.02 = 0.98\r<BR>\n" );
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document.write( "The four pieces of useful relevant info are<BR>\n" );
document.write( "P(A) = 0.57<BR>\n" );
document.write( "P(B) = 0.43<BR>\n" );
document.write( "P(~D given A) = 0.94<BR>\n" );
document.write( "P(~D given B) = 0.98\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "P(~D) = P(~D and A) + P(~D and B) ........ Law of total probability<BR>\n" );
document.write( "P(~D) = P(~D given A)*P(A) + P(~D given B)*P(B)<BR>\n" );
document.write( "P(~D) = 0.94*0.57 + 0.98*0.43<BR>\n" );
document.write( "P(~D) = <font color=red size=4>0.9572</font> which is the final answer.<BR>\n" );
document.write( "There's a 95.72% chance the part is not defective.\r<BR>\n" );
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document.write( "An alternative approach:\r<BR>\n" );
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document.write( "Let's say that 100,000 parts were made.<BR>\n" );
document.write( "Machine A makes 57,000 of those parts (because 57% of 100,000 = 57,000). The remaining 100,000-57,000 = 43,000 parts are from machine B.<BR>\n" );
document.write( "Note how 43% of 100,000 is 43,000.\r<BR>\n" );
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document.write( "Machine A messes up 6% of the time.<BR>\n" );
document.write( "6% of 57,000 = 0.06*57000 = 3420<BR>\n" );
document.write( "There are 3420 defective parts made from machine A. The remaining 57,000-3,420 = 53,580 parts are not defective.<BR>\n" );
document.write( "Or we could say: 94% of 57,000 = 0.94*57000 = 53,580.\r<BR>\n" );
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document.write( "Machine B messes up 2% of the time.<BR>\n" );
document.write( "2% of 43,000 = 0.02*43000 = 860<BR>\n" );
document.write( "There are 860 defective parts made from machine B. The remaining 43,000-860 = 42,140 parts are not defective.<BR>\n" );
document.write( "Or we could say: 98% of 43,000 = 0.98*43000 = 42,140.\r<BR>\n" );
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document.write( "In total, there are 53580+42140 = 95,720 non-defective parts out of 100,000 parts total.\r<BR>\n" );
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document.write( "Here's a table to summarize everything stated so far in this section.<BR>\n" );
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document.write( "<tr><TD ></TD><TD >Defective</TD><TD >Not Defective</TD><TD >Total</TD></tr><tr><TD >Machine A</TD><TD >3,420</TD><TD >53,580</TD><TD >57,000</TD></tr><tr><TD >Machine B</TD><TD >860</TD><TD >42,140</TD><TD >43,000</TD></tr><tr><TD >Total</TD><TD >4,280</TD><TD >95,720</TD><TD >100,000</TD></tr>\n" );
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document.write( "The probability of randomly selecting a non-defective part is (95,720)/(100,000) = <font color=red size=4>0.9572</font>\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>0.9572</font><BR>\n" );
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