document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1030185>Question 1030185</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a lottery game, a player picks six numbers from 1 to 24. If the player matches all six numbers, they win $ 20,000. Otherwise, they lose $1. Assume it costs nothing to play.\r<BR>\n" );
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document.write( "What is the expected value of this game?\r<BR>\n" );
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document.write( "I'm not sure where to even start with this one. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #645099 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=645099\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Question:<BR>\n" );
document.write( "In a lottery game, a player picks six numbers from 1 to 24. If the player matches all six numbers, they win $ 20,000. Otherwise, they lose $1. Assume it costs nothing to play.<BR>\n" );
document.write( "What is the expected value of this game?<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "To start solving the problem, we need to understand the definitions of words, the most important of which is \"expected value\".<BR>\n" );
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document.write( "Wiki defines expected value as:<BR>\n" );
document.write( "\"The expected value is also known as the expectation, mathematical expectation, EV, average, mean value, mean, or first moment. More practically, the expected value of a discrete random variable is the probability-weighted average of all possible values.\"<BR>\n" );
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document.write( "To mathematically calculate the expected value, we will need to do the sum of the product<BR>\n" );
document.write( "x*P(x)<BR>\n" );
document.write( "for all possible (outcome) values of x, that is, all values of x for which the probability is non-zero.<BR>\n" );
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document.write( "Here the possible outcomes are x=-1 (lose) or x=20000 (win)<BR>\n" );
document.write( "Total possible number of outcomes = C(24,6) [24 choose 6]<BR>\n" );
document.write( "=24!/(6!18!)<BR>\n" );
document.write( "= 134596<BR>\n" );
document.write( "Out of which there is only one winning combination.<BR>\n" );
document.write( "Therefore we conclude:<BR>\n" );
document.write( "P(win 20000)=1/134596<BR>\n" );
document.write( "P(lose 1)=134595/134596<BR>\n" );
document.write( "and hence the expected value is:<BR>\n" );
document.write( "20000*(1/134596)+(-1)*(134595/134596)<BR>\n" );
document.write( "=-114595/134596<BR>\n" );
document.write( "=-0.8514 (rounded to four places after decimal)\r<BR>\n" );
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document.write( "Interpretation:<BR>\n" );
document.write( "In the long run, if the game is played many, many times, the <i>average</i> loss is $0.8514 per play.<BR>\n" );
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