document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1010946>Question 1010946</A>: <I><SPAN STYLE=\"word-break: break-all;\"> John purchases an office complex that contains 150 offices. Records show that in a recession year only 70% of the offices are rented. John needs at least 110 offices to be rented to breakeven. John has asked you to determine, what is the probability John will breakeven this recession?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #626498 by </B><A HREF=/tutors/aboutme.mpl?userid=mathmate><B>mathmate(429)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathmate><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=mathmate><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=626498\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Question:<BR>\n" );
document.write( "John purchases an office complex that contains 150 offices. Records show that in a recession year only 70% of the offices are rented. John needs at least 110 offices to be rented to breakeven. John has asked you to determine, what is the probability John will breakeven this recession? \r<BR>\n" );
document.write( "\n" );
document.write( "Solution:<BR>\n" );
document.write( "This problem can be solved using the binomial distribution, since probability of renting remains constant at 70% throughout, and assuming independence of rentals.  Also the number of trials (150) is known.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Parameters of the distribution are:<BR>\n" );
document.write( "n=150, p=0.7 <BR>\n" );
document.write( "we look for<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X%3E=110%29=sum%28C%28n%2Cx%29%2Ap%5Ex%2A%281-p%29%5E%28n-x%29%29 ALIGN=MIDDLE  ALT=\"P%28X%3E=110%29=sum%28C%28n%2Cx%29%2Ap%5Ex%2A%281-p%29%5E%28n-x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for x=110 to 150<BR>\n" );
document.write( "The summation may be carried out to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X%3E=110%29=0.21261 ALIGN=MIDDLE  ALT=\"P%28X%3E=110%29=0.21261\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "In practice, it is usual to use the normal approximation with continuity correction:<BR>\n" );
document.write( "P(X>=110) is approximated by calculating the equivalent normal parameters,<BR>\n" );
document.write( "mean = np=105<BR>\n" );
document.write( "variance = npq=31.5<BR>\n" );
document.write( "standard deviation = sqrt(npq)=5.6125<BR>\n" );
document.write( "Therefore, using normal approximation,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X%3E=109.5%29=P%28Z%3E=%28109.5-105%29%2F5.6125%29=P%28Z%3E=0.80178%29=%281-0.78866%29=0.21134 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Answer:<BR>\n" );
document.write( "Using binomial distribution <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X%3E=110%29=0.21261 ALIGN=MIDDLE  ALT=\"P%28X%3E=110%29=0.21261\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "using normal approximation, with continuity correction,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X%3E=110%29=0.21134 ALIGN=MIDDLE  ALT=\"P%28X%3E=110%29=0.21134\"   DISABLED_style= \"max-width:90%; height:auto;\" > (approx.) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );