document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1090959>Question 1090959</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A car dealership has 6 &#8203;red, 14 &#8203;silver, and 5 black cars on the lot. Ten cars are randomly chosen to be displayed in front of the dealership. Complete parts&#8203; (a) through&#8203; (c) below.\r<BR>\n" );
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document.write( " Find the probability that 4 cars are red and the rest are silver.\r<BR>\n" );
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document.write( "nothing<BR>\n" );
document.write( "&#8203;(Round to four decimal places as&#8203; needed.) <BR>\n" );
document.write( "&#8203;(b) Find the probability that <BR>\n" );
document.write( "5<BR>\n" );
document.write( "5 cars are red and <BR>\n" );
document.write( "5<BR>\n" );
document.write( "5 are black.\r<BR>\n" );
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document.write( "nothing<BR>\n" );
document.write( "&#8203;(Round to six decimal places as&#8203; needed.) <BR>\n" );
document.write( "&#8203;(c) Find the probability that exactly 6 cars are red.\r<BR>\n" );
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document.write( "nothing<BR>\n" );
document.write( "&#8203;(Round to five decimal places as&#8203; needed.)  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #705404 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=705404\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Question: (please check if edited version is the same as original question)<BR>\n" );
document.write( "A car dealership has 6 &#8203;red, 14 &#8203;silver, and 5 black cars on the lot. Ten cars are randomly chosen to be displayed in front of the dealership. Complete parts&#8203; (a) through&#8203; (c) below.<BR>\n" );
document.write( "(a) Find the probability that 4 cars are red and the rest are silver.<BR>\n" );
document.write( "    (Round to four decimal places as&#8203; needed.) <BR>\n" );
document.write( "(b) Find the probability that 5 cars are red and 5 are black.<BR>\n" );
document.write( "    (Round to six decimal places as&#8203; needed.) <BR>\n" );
document.write( "(c) Find the probability that exactly 6 cars are red.<BR>\n" );
document.write( "    (Round to five decimal places as&#8203; needed.) <BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "This is a hypergeometric distribution problem.<BR>\n" );
document.write( "When we have A objects of one class, and B objects of another.  The probability of picking a from the A class and b from the B class is given by:<BR>\n" );
document.write( "P(a,b)=C(A,a)*C(B,b)/C(A+B,a+b)<BR>\n" );
document.write( "where C(n,r) is the number of combinations of picking r objects from n, and where<BR>\n" );
document.write( "C(n,r)=n!/(r!(n-r)!)<BR>\n" );
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document.write( "The stock to choose from is 6 red, 14 silver and 5 black, for a total of 25 cars.<BR>\n" );
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document.write( "(a) 4 red and 6 silver <BR>\n" );
document.write( "Since the hypergeometric distribution formula above applies to only two classes, we can apply the formula in two steps:<BR>\n" );
document.write( "Step 1: assume silver or red together (20 cars) as one class, and black as another.  Choose 10 from the combined class and calculate probability using hypergeometric distribution.<BR>\n" );
document.write( "P(10 red or silver)=C(20,10)*C(5,0)/C(25,10)<BR>\n" );
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document.write( "Step 2: choose 4 red and 6 black from the 10 cars chosen, again using the hypergeometric distribution.<BR>\n" );
document.write( "P(4 red + 6 silver from 20)=C(6,4)*C(14,6)/C(20,10)<BR>\n" );
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document.write( "We then multiply the two probabilities together to get<BR>\n" );
document.write( "P(4 red + 6 silver from stock of 25)<BR>\n" );
document.write( "=C(20,10)*C(5,0)/C(25,10) * C(6,4)*C(14,6)/C(20,10)<BR>\n" );
document.write( "=C(6,4)*C(14,6)*C(5,0)/C(25,10)  after simplification<BR>\n" );
document.write( "which is exactly the multinomial equivalent of the hypergeometric distribution for 4 red, 6 silver and 0 black.<BR>\n" );
document.write( "Evaluating,<BR>\n" );
document.write( "P(4 red+6 silver)<BR>\n" );
document.write( "=C(6,4)*C(14,6)*C(5,0)/C(25,10)<BR>\n" );
document.write( "=15*3003*1/3268760<BR>\n" );
document.write( "=819/59432<BR>\n" );
document.write( "=0.013780<BR>\n" );
document.write( " <BR>\n" );
document.write( "(b) 5 red + 5 black<BR>\n" );
document.write( "Apply the multinomial hypergeometric distribution:<BR>\n" );
document.write( "P(5 red + 5 black)<BR>\n" );
document.write( "=C(6,5)*C(14,0)*C(5,5)/C(25,10)<BR>\n" );
document.write( "=6*1*1/3268760<BR>\n" );
document.write( "=3/1634380<BR>\n" );
document.write( "=0.00000183556<BR>\n" );
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document.write( "(c) exactly 6 cars are red<BR>\n" );
document.write( "We can apply directly hypergeometric distribution by putting red as one class, and silver or black (total 14+5=19) as the other.<BR>\n" );
document.write( "P(6 red + 4 (silver or black)<BR>\n" );
document.write( "=C(6,6)*C(19,4)/C(25,10)<BR>\n" );
document.write( "=1*3876/3268780<BR>\n" );
document.write( "=0.001186<BR>\n" );
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