document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1155746>Question 1155746</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A mortgage company classifies its borrowers into three categories: Low Risk, Medium Risk, and High Risk. From experience, the company knows that:\r<BR>\n" );
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document.write( "3% of low risk borrowers eventually default on their mortgages.<BR>\n" );
document.write( "7% of medium risk borrowers eventually default on their mortgages.<BR>\n" );
document.write( "13% of high risk borrowers eventually default on their mortgages.<BR>\n" );
document.write( "The mortgages for 133 high risk borrowers are put together into one portfolio. The company determines that they will profit on the portfolio as long as no more than 19% of borrowers with mortgages in the portfolio default. What is the probability that the company makes a profit on the portfolio? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #778385 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=778385\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 19% of 133 is 25.27, so want no more than 25 to default. <BR>\n" );
document.write( "On calculator use binomcdf (133,.13,26) and probability is 0.9879. ANSWER\r<BR>\n" );
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document.write( "Can check by using probability of 26 defaulting or 133C26*.13^26*.87^107=0.0093, then probability of 27 and 28 until the numbers are too small.  But given what 26 is, the other ones will be smaller and the answer is reasonable.  \r<BR>\n" );
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document.write( "Normal approximation is np=17.29;  np(1-p)^1/2 is sd =3.88<BR>\n" );
document.write( "want probability it is less than 25.<BR>\n" );
document.write( "z=(25-17.29)/3.88<BR>\n" );
document.write( "=1.99<BR>\n" );
document.write( "Probability z>1.99=0.0232<BR>\n" );
document.write( "probability z<1.99=0.9768, with error (this is an approximation) but again close to exact answer above.\r<BR>\n" );
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