document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151739>Question 1151739</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a state lottery a seven-digit integer is selected at random. If a player<BR>\n" );
document.write( "bets 1 dollar on a particular number, the payoff (if that number is selected)<BR>\n" );
document.write( "is $600 minus the $1 paid for the ticket. Let X equal the payoff to the better.<BR>\n" );
document.write( "Find the expected value of X. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #773569 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=773569\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Let's say we had 7 slots A,B,C,D,E,F,G<BR>\n" );
document.write( "The slots will represent the digits of the 7-digit number.\r<BR>\n" );
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document.write( "We have 9 choices to fill for slot A. We can choose anything from {1,2,3,4,5,6,7,8,9}<BR>\n" );
document.write( "We have 10 choices to fill for slot B. We can choose anything from {0,1,2,3,4,5,6,7,8,9}<BR>\n" );
document.write( "Slots C through G will have the same number of choices as slot B.\r<BR>\n" );
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document.write( "Here are the number of choices we have for slots A through G<BR>\n" );
document.write( "9, 10, 10, 10, 10, 10, 10\r<BR>\n" );
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document.write( "There are 9*10*10*10*10*10*10 = 9,000,000 different seven-digit numbers<BR>\n" );
document.write( "The smallest of which is 1,000,000 and the largest is 9,999,999<BR>\n" );
document.write( "This figure of 9,000,000 can also be found by subtracting the endpoints (1,000,000 and 9,999,999) and then adding 1<BR>\n" );
document.write( "9,999,999 - 1,000,000 + 1 = 8,999,999 + 1 = 9,000,000\r<BR>\n" );
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document.write( "Whichever method you use, there are 9,000,000 different seven-digit numbers\r<BR>\n" );
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document.write( "There is only one winning ticket\r<BR>\n" );
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document.write( "The probability of winning is <BR>\n" );
document.write( "1/(9,000,000) = 0.000000111 approximately\r<BR>\n" );
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document.write( "The probability of losing is<BR>\n" );
document.write( "1-0.000000111 = 0.999999889 approximately\r<BR>\n" );
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document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
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document.write( "<tr><TD >Outcome</TD><TD >Net Payoff X</TD><TD >P(X)</TD><TD >X*P(X)</TD></tr>\r\n" );
document.write( "<tr><TD >Win</TD><TD >599</TD><TD >0.000000111</TD><TD >0.000066489</TD></tr>\r\n" );
document.write( "<tr><TD >Lose</TD><TD >-1</TD><TD >0.999999889</TD><TD >-0.999999889</TD></tr>\r\n" );
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document.write( "</TABLE><BR>\n" );
document.write( "Adding up the X*P(X) values gets us<BR>\n" );
document.write( "0.000066489 + (-0.999999889) = -0.9999334<BR>\n" );
document.write( "which rounds to -1.00\r<BR>\n" );
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document.write( "On average, the person playing the lottery will lose $1<BR>\n" );
document.write( "This is effectively the same as saying \"on average, the person playing the lottery will not win the jackpot\" since the price to play the game is $1.<BR>\n" );
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