document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197325>Question 1197325</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Five of 10 new buildings in a city violate the building code. A random sample of three buildings<BR>\n" );
document.write( "selected for inspection.<BR>\n" );
document.write( "a)What is the probability of none of the buildings violate the building code in a sample of 3 buildings\r<BR>\n" );
document.write( "\n" );
document.write( "b)What is the probability of at least one of the new buildings violate the building code in a sample of 3 buildings? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830556 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=830556\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5 buildings violate code, 10-5 = 5 do not\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5/10 = probability first selection doesn't violate code<BR>\n" );
document.write( "4/9 = probability second selection doesn't violate code<BR>\n" );
document.write( "3/8 = probability third selection doesn't violate code\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The numerators decrease: 5,4,3<BR>\n" );
document.write( "So do the denominators: 10,9,8<BR>\n" );
document.write( "This assumes that we're sampling without replacement.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Multiply out the fractions to get this result<BR>\n" );
document.write( "(5/10)*(4/9)*(3/8) = (5*4*3)/(10*9*8)<BR>\n" );
document.write( "(5/10)*(4/9)*(3/8) = (5*4*3)/(2*5*3*3*2*4)<BR>\n" );
document.write( "(5/10)*(4/9)*(3/8) = 1/(2*3*2)<BR>\n" );
document.write( "(5/10)*(4/9)*(3/8) = 1/12\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The probability of selecting 3 buildings that don't violate code is 1/12.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Alternate route:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "n = 5 buildings that don't violate code<BR>\n" );
document.write( "r = 3 selections<BR>\n" );
document.write( "Use the nCr combination formula<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "5 C 3 = (5!)/(3!*(5-3)!)<BR>\n" );
document.write( "5 C 3 = (5!)/(3!*2!)<BR>\n" );
document.write( "5 C 3 = (5*4*3!)/(3!*2!)<BR>\n" );
document.write( "5 C 3 = (5*4)/(2!)<BR>\n" );
document.write( "5 C 3 = (5*4)/(2*1)<BR>\n" );
document.write( "5 C 3 = (20)/(2)<BR>\n" );
document.write( "5 C 3 = 10<BR>\n" );
document.write( "There are 10 ways to pick three out of the five buildings that are to code.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Using that same formula, you should find that 10C3 = 120 which represents the number of ways to pick any three buildings (whether they are to code or not).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "As you can probably see, we are using nCr instead of nPr because order doesn't matter. A grouping like ABC is the same as CBA.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We have 10 ways to pick the three buildings that are to code out of 120 ways to pick the three buildings.<BR>\n" );
document.write( "10/120 = 1/12\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: <font color=red>1/12</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "======================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Part (b)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We'll use the answer of the previous part.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The events \"all 3 don't violate code\" and \"at least one violates code\" are complementary events. <BR>\n" );
document.write( "One or the other must happen. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "P(all 3 don't violate code) + P(at least one violates code) = 1<BR>\n" );
document.write( "P(at least one violates code) = 1 - P(all 3 don't violate code)<BR>\n" );
document.write( "P(at least one violates code) = 1 - 1/12<BR>\n" );
document.write( "P(at least one violates code) = 12/12 - 1/12<BR>\n" );
document.write( "P(at least one violates code) = (12 - 1)/12<BR>\n" );
document.write( "P(at least one violates code) = 11/12\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: <font color=red>11/12</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );