document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1029767>Question 1029767</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Could you please help me with this question?\r<BR>\n" );
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document.write( "The door prizes at a dance are four $10 gift certificates, five $20 gift certificates, and three $50 gift certificates. The prize envelopes are mixed together in a bag, and five prizes are drawn at random. \r<BR>\n" );
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document.write( "What is the probability that none of the prizes is a $10 gift certificate? \r<BR>\n" );
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document.write( "What is the expected number of $20 gift certificates drawn?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #645197 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=645197\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Question:The door prizes at a dance are four $10 gift certificates, five $20 gift certificates, and three $50 gift certificates. The prize envelopes are mixed together in a bag, and five prizes are drawn at random. <BR>\n" );
document.write( "(a) What is the probability that none of the prizes is a $10 gift certificate? <BR>\n" );
document.write( "(b) What is the expected number of $20 gift certificates drawn?<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "This is a typical problem of picking objects from a mixture of \"good\" and \"bad\" objects without replacement.  The modelling distribution is the hypergeometric distribution, given by:<BR>\n" );
document.write( "P(a)=C(A,a)*C(B,b)/C(A+B,a+b)<BR>\n" );
document.write( "where <BR>\n" );
document.write( "a=number of \"good\" objects picked<BR>\n" );
document.write( "b=number of \"bad\" objects picked.<BR>\n" );
document.write( "A=number of \"good\" objects in the given batch, and<BR>\n" );
document.write( "B=number of \"bad\" objects in the given batch.<BR>\n" );
document.write( "a+b=total number of objects picked (without replacement)<BR>\n" );
document.write( "A+B=total number of objects in the batch.<BR>\n" );
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document.write( "(a) P($10=0<BR>\n" );
document.write( "Here A=4 ($10, good), B=3+5=8 ($20, $50, \"bad\")<BR>\n" );
document.write( "a=0, b=5 (remember, a+b=5)<BR>\n" );
document.write( "therefore<BR>\n" );
document.write( "P(a=0)=C(4,0)*C(8,5)/C(12,5)<BR>\n" );
document.write( "=1*56/792<BR>\n" );
document.write( "=7/99<BR>\n" );
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document.write( "(b) Find E[$20]<BR>\n" );
document.write( "$20 is the \"good\" prize.<BR>\n" );
document.write( "A=5, B=3+4=7, a+b=5 (total number of prizes drawn)<BR>\n" );
document.write( "E[$20]=(5/(5+3+4)*5)=(5/12)*5=25/12=2.083 (to 3 decimal places)<BR>\n" );
document.write( "Expected value is equation to (A/(A+B))*(a+b)\r<BR>\n" );
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