document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169525>Question 1169525</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A previous survey shows that a machine making plastic components is correctly set<BR>\n" );
document.write( "up for the day's production on 85% of days. On days when it is set up correctly, 96%<BR>\n" );
document.write( "of the components produced are good. If the machine is not set up correctly, only 30%<BR>\n" );
document.write( "of the components produced are good. On a particular day, the machine is set up and<BR>\n" );
document.write( "the first component produced is found to be good. What is the probability that the<BR>\n" );
document.write( "machine is set up correctly? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851314 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.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=851314\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem using Bayes' Theorem.\r<BR>\n" );
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document.write( "**Define the Events**\r<BR>\n" );
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document.write( "* **C:** The machine is set up correctly.<BR>\n" );
document.write( "* **G:** The first component produced is good.\r<BR>\n" );
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document.write( "**Given Probabilities**\r<BR>\n" );
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document.write( "* $P(C) = 0.85$ (Probability the machine is set up correctly)<BR>\n" );
document.write( "* $P(C') = 1 - P(C) = 1 - 0.85 = 0.15$ (Probability the machine is not set up correctly)<BR>\n" );
document.write( "* $P(G|C) = 0.96$ (Probability of a good component given the machine is set up correctly)<BR>\n" );
document.write( "* $P(G|C') = 0.30$ (Probability of a good component given the machine is not set up correctly)\r<BR>\n" );
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document.write( "**We need to find:**\r<BR>\n" );
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document.write( "* $P(C|G)$ (Probability the machine is set up correctly given the component is good)\r<BR>\n" );
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document.write( "**Apply Bayes' Theorem**\r<BR>\n" );
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document.write( "Bayes' Theorem states:\r<BR>\n" );
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document.write( "$$P(C|G) = \frac{P(G|C) \cdot P(C)}{P(G)}$$\r<BR>\n" );
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document.write( "Where $P(G)$ can be found using the law of total probability:\r<BR>\n" );
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document.write( "$$P(G) = P(G|C) \cdot P(C) + P(G|C') \cdot P(C')$$\r<BR>\n" );
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document.write( "**Calculations**\r<BR>\n" );
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document.write( "1.  **Calculate $P(G)$:**<BR>\n" );
document.write( "    $$P(G) = (0.96 \times 0.85) + (0.30 \times 0.15)$$<BR>\n" );
document.write( "    $$P(G) = 0.816 + 0.045$$<BR>\n" );
document.write( "    $$P(G) = 0.861$$\r<BR>\n" );
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document.write( "2.  **Calculate $P(C|G)$:**<BR>\n" );
document.write( "    $$P(C|G) = \frac{P(G|C) \cdot P(C)}{P(G)}$$<BR>\n" );
document.write( "    $$P(C|G) = \frac{0.96 \times 0.85}{0.861}$$<BR>\n" );
document.write( "    $$P(C|G) = \frac{0.816}{0.861}$$<BR>\n" );
document.write( "    $$P(C|G) \approx 0.9477$$\r<BR>\n" );
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document.write( "**Final Answer**\r<BR>\n" );
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document.write( "The probability that the machine is set up correctly given that the first component produced is good is approximately 0.9477.<BR>\n" );
document.write( " </SPAN>\n" );
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