document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181553>Question 1181553</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A company buys 20% of its electronic chips from supplier New and the rests from other suppliers. Assume that the company purchases a large number of chips from each supplier.<BR>\n" );
document.write( "It is known that 1/10 (one tenth) of supplier New’s chips are defective while the other suppliers have only 1/50 (one fiftieth) chips defective. The company selects 20 chips at random from a batch and tests them. One chip is found to be defective.<BR>\n" );
document.write( "What is the probability that this batch of chips came from supplier New?<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 #849994 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=849994\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem using Bayes' Theorem:\r<BR>\n" );
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document.write( "**1. Define Events:**\r<BR>\n" );
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document.write( "*   N: The batch of chips came from supplier New.<BR>\n" );
document.write( "*   O: The batch of chips came from other suppliers.<BR>\n" );
document.write( "*   D: A randomly selected chip is defective.\r<BR>\n" );
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document.write( "**2. Given Probabilities:**\r<BR>\n" );
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document.write( "*   P(N) = 0.20 (20% of chips are from New)<BR>\n" );
document.write( "*   P(O) = 1 - P(N) = 0.80 (80% of chips are from other suppliers)<BR>\n" );
document.write( "*   P(D|N) = 1/10 = 0.10 (Probability of a defective chip given it's from New)<BR>\n" );
document.write( "*   P(D|O) = 1/50 = 0.02 (Probability of a defective chip given it's from other suppliers)\r<BR>\n" );
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document.write( "**3. What We Want:**\r<BR>\n" );
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document.write( "We want to find P(N|D), the probability that the batch came from supplier New *given* that one chip is found to be defective.\r<BR>\n" );
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document.write( "**4. Bayes' Theorem:**\r<BR>\n" );
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document.write( "Bayes' Theorem states:\r<BR>\n" );
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document.write( "P(N|D) = [P(D|N) * P(N)] / P(D)\r<BR>\n" );
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document.write( "We need to find P(D), the overall probability of a defective chip.  We can use the law of total probability:\r<BR>\n" );
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document.write( "P(D) = P(D|N) * P(N) + P(D|O) * P(O)<BR>\n" );
document.write( "P(D) = (0.10 * 0.20) + (0.02 * 0.80)<BR>\n" );
document.write( "P(D) = 0.02 + 0.016<BR>\n" );
document.write( "P(D) = 0.036\r<BR>\n" );
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document.write( "**5. Apply Bayes' Theorem:**\r<BR>\n" );
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document.write( "P(N|D) = (0.10 * 0.20) / 0.036<BR>\n" );
document.write( "P(N|D) = 0.02 / 0.036<BR>\n" );
document.write( "P(N|D) ≈ 0.556\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "The probability that the batch of chips came from supplier New, given that one chip is defective, is approximately 0.556 or 55.6%.<BR>\n" );
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