document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=365423>Question 365423</A>: <I><SPAN STYLE=\"word-break: break-all;\"> One thousand raffle tickets are sold for $5.00 each\r<BR>\n" );
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document.write( "One grand prize of $800 and two consolation prizes of $100 each will be awarded. Jeremy purchases one ticket. Find his expected value </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #260538 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=260538\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1,000 tickets sold for 5.00 apiece to bring in 5,000 dollars.<BR>\n" );
document.write( "1 grand prize of 800 and 2 consolation prizes of 100 each will be awarded.<BR>\n" );
document.write( "jeremy purchases one ticket.<BR>\n" );
document.write( "what is his expected value.\r<BR>\n" );
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document.write( "he has 1 out of 1000 chances to win 800.<BR>\n" );
document.write( "he has 2 out of 1000 chances to win 100.\r<BR>\n" );
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document.write( "the expected value is the sum of the net value of each option times the probability that the option will be granted.\r<BR>\n" );
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document.write( "in his case:\r<BR>\n" );
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document.write( "he spends 5.00 to buy a ticket.<BR>\n" );
document.write( "if he wins the pot, then he will receive 800 dollars.<BR>\n" );
document.write( "his net value is $800 - $5 = $775 dollars.\r<BR>\n" );
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document.write( "if he wins one of the second prizes then he will receive 100 dollars.<BR>\n" );
document.write( "his net value is $100 - $5 = $95 dollars.\r<BR>\n" );
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document.write( "this can happen 2 times.\r<BR>\n" );
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document.write( "if he doesn't win anything, then his net value is -$5.00\r<BR>\n" );
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document.write( "the probability that he will win first prize is 1/1000.<BR>\n" );
document.write( "the probability that he will win one of the second prizes is 1/1000.<BR>\n" );
document.write( "the probability that he will win the other of the second prizes is 1/1000<BR>\n" );
document.write( "the probability that he will not win any of the prizes is 997/1000.\r<BR>\n" );
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document.write( "The general formula for expected value is:\r<BR>\n" );
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document.write( "a1 * p(a1) + a2 * p(a2) + a3 * p(a3) + .....aN * p(aN).\r<BR>\n" );
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document.write( "a1 through aN are the net values for each outcome.<BR>\n" );
document.write( "p(a1) through p(aN) are the probabilities for each outcome.\r<BR>\n" );
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document.write( "for jeremy's problem, the expected value is:\r<BR>\n" );
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document.write( "795 * (1/1000) + 95 * (1/1000) + (95/1/1000) + (-5)*(997/1000)\r<BR>\n" );
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document.write( "the net result of all of this is that his expected value is -$4.00 dollars.\r<BR>\n" );
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document.write( "Looking at this another way, we should be able to come up with the same answers.\r<BR>\n" );
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document.write( "he spends $5.00 each time.\r<BR>\n" );
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document.write( "1% of the time he will win $800.<BR>\n" );
document.write( "2% of the time he will win $100.<BR>\n" );
document.write( "97% of the time he will win $0.00.\r<BR>\n" );
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document.write( "If he plays the game 1000 times, then:\r<BR>\n" );
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document.write( "he will spend $5.00 1000 times for a total of $5,000.00<BR>\n" );
document.write( "he will win $800.00 1 time for a total of $800.00<BR>\n" );
document.write( "he will win $100.00 2 times for a total of $200.00<BR>\n" );
document.write( "he will win nothing 997 times for a total of $0.00\r<BR>\n" );
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document.write( "His total net value is $1000 - $5000 = -$4000.\r<BR>\n" );
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document.write( "divide that by 1000 and his average net value for each game is -$4.00.\r<BR>\n" );
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document.write( "This is the same as the answser we derived from the expected value formula.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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