document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1086242>Question 1086242</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A local lottery is selling scratch-off tickets for $3.00 each.  There are nine scratch-off spots consisting of the following dollar amounts:  Four $5 spots, three $10 spots, and two $20 spots. If you scratch only two and get a matching pair you win the dollar amount listed ($5, $10, or $20).<BR>\n" );
document.write( "a.	What is the probability that you will NOT win any prize?\r<BR>\n" );
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document.write( "b.	What is the probability that you will win $20?\r<BR>\n" );
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document.write( "c.	Find your expected gain or loss.<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 #700397 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=700397\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Question:A local lottery is selling scratch-off tickets for $3.00 each.  There are nine scratch-off spots consisting of the following dollar amounts:  Four $5 spots, three $10 spots, and two $20 spots. If you scratch only two and get a matching pair you win the dollar amount listed ($5, $10, or $20).<BR>\n" );
document.write( "a.	What is the probability that you will NOT win any prize?<BR>\n" );
document.write( "b.	What is the probability that you will win $20?<BR>\n" );
document.write( "c.	Find your expected gain or loss.<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "Expected value is calculated by summing, for all possible outcomes,<BR>\n" );
document.write( "&sum;pi(Xi)<BR>\n" );
document.write( "where<BR>\n" );
document.write( "pi = probability of outcome i<BR>\n" );
document.write( "Xi = revenue (gain-cost) of outcome i.<BR>\n" );
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document.write( "For the given scratch-off sale, customers pay $3 whether they win or not.<BR>\n" );
document.write( "The spots are:<BR>\n" );
document.write( "$5  (4)<BR>\n" );
document.write( "$10 (3)<BR>\n" );
document.write( "$20 (2)<BR>\n" );
document.write( "By scratching 2 and match, customer get the amount (less $3 already paid).<BR>\n" );
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document.write( "P($5)=4/9*3/8=1/6<BR>\n" );
document.write( "P($10)=3/9*2/8=1/12<BR>\n" );
document.write( "P(20)=2/9*1/8=1/36<BR>\n" );
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document.write( "(a) probability of winning nothing (i.e. lose $3)<BR>\n" );
document.write( "P(0)=1-(1/6+1/12+1/36)=1-5/18=13/18<BR>\n" );
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document.write( "(b)P($20)=1/36 (see above)<BR>\n" );
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document.write( "(c) E[X]<BR>\n" );
document.write( "=(5-3)P($5)+(10-3)P($10)+(20-3)P($20)+(0-3)P(0)<BR>\n" );
document.write( "=2(1/6)+7(1/12)+17(1/36)+(-3)(13/18)<BR>\n" );
document.write( "= -7/9 =-0.78 (approx.)<BR>\n" );
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