document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176518>Question 1176518</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a consulting firm has submitted a bid for a large research project . The firm's management initially felt that there was a 50-50 chance of getting the bid . However , the agency to which the bid was submitted has subsequently requested additional information on the bid . Past experience indicates that on 75 % of the successful bids and 40 % of the unsuccessful bids additional information is requested ( a ) What is the probability that the bid will be successful ? ( ( b ) What is the conditional probability of a request for additional information , given that the bid will ultimately be successful ? (c)What is the probability that the agency will ask for the additional information ?( d ) Compute the probability that the bid will be successful given that a request for additional information has been received ?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850574 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=850574\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem using Bayes' Theorem.\r<BR>\n" );
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document.write( "**Define Events:**\r<BR>\n" );
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document.write( "* S: The bid is successful.<BR>\n" );
document.write( "* U: The bid is unsuccessful.<BR>\n" );
document.write( "* R: Additional information is requested.\r<BR>\n" );
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document.write( "**Given Probabilities:**\r<BR>\n" );
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document.write( "* P(S) = 0.50 (Initial probability of success)<BR>\n" );
document.write( "* P(U) = 0.50 (Initial probability of failure)<BR>\n" );
document.write( "* P(R|S) = 0.75 (Probability of request given success)<BR>\n" );
document.write( "* P(R|U) = 0.40 (Probability of request given failure)\r<BR>\n" );
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document.write( "**(a) What is the probability that the bid will be successful?**\r<BR>\n" );
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document.write( "This is simply the initial given probability:\r<BR>\n" );
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document.write( "* P(S) = 0.50\r<BR>\n" );
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document.write( "**(b) What is the conditional probability of a request for additional information, given that the bid will ultimately be successful?**\r<BR>\n" );
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document.write( "This is directly given:\r<BR>\n" );
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document.write( "* P(R|S) = 0.75\r<BR>\n" );
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document.write( "**(c) What is the probability that the agency will ask for the additional information?**\r<BR>\n" );
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document.write( "We need to use the law of total probability:\r<BR>\n" );
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document.write( "* P(R) = P(R|S) * P(S) + P(R|U) * P(U)<BR>\n" );
document.write( "* P(R) = (0.75 * 0.50) + (0.40 * 0.50)<BR>\n" );
document.write( "* P(R) = 0.375 + 0.20<BR>\n" );
document.write( "* P(R) = 0.575\r<BR>\n" );
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document.write( "**(d) Compute the probability that the bid will be successful given that a request for additional information has been received?**\r<BR>\n" );
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document.write( "We need to use Bayes' Theorem:\r<BR>\n" );
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document.write( "* P(S|R) = [P(R|S) * P(S)] / P(R)<BR>\n" );
document.write( "* P(S|R) = (0.75 * 0.50) / 0.575<BR>\n" );
document.write( "* P(S|R) = 0.375 / 0.575<BR>\n" );
document.write( "* P(S|R) ≈ 0.6522\r<BR>\n" );
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document.write( "**Answers:**\r<BR>\n" );
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document.write( "* **(a) P(S) = 0.50**<BR>\n" );
document.write( "* **(b) P(R|S) = 0.75**<BR>\n" );
document.write( "* **(c) P(R) = 0.575**<BR>\n" );
document.write( "* **(d) P(S|R) ≈ 0.6522**<BR>\n" );
document.write( " </SPAN>\n" );
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