document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200826>Question 1200826</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose someone was playing the following dice game. The game costs $5 to play. You roll two dice and take the sum. If the sum is 2 or 12, you get $10. If the sum is 3 or 11, you get $9. If the sum is 4 or 10, you get $8. If the sum is 5 or 9, you get $5 (you get your money back). If the sum is 6, 7, or 8, you get nothing.\r<BR>\n" );
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document.write( "a. What is the expected value of this game, taking into consideration the $5 cost of playing the game? What does this mean?\r<BR>\n" );
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document.write( "b. Does the game favor you, or the people who play it? </SPAN>\n" );
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document.write( "Part (a)<BR>\n" );
document.write( "Refer to this sum of dice chart.<BR>\n" );
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document.write( "<tr><TD >+</TD><TD ><font color=red>1</font></TD><TD ><font color=red>2</font></TD><TD ><font color=red>3</font></TD><TD ><font color=red>4</font></TD><TD ><font color=red>5</font></TD><TD ><font color=red>6</font></TD></tr><tr><TD ><font color=blue>1</font></TD><TD >2</TD><TD >3</TD><TD >4</TD><TD >5</TD><TD >6</TD><TD >7</TD></tr><tr><TD ><font color=blue>2</font></TD><TD >3</TD><TD >4</TD><TD >5</TD><TD >6</TD><TD >7</TD><TD >8</TD></tr><tr><TD ><font color=blue>3</font></TD><TD >4</TD><TD >5</TD><TD >6</TD><TD >7</TD><TD >8</TD><TD >9</TD></tr><tr><TD ><font color=blue>4</font></TD><TD >5</TD><TD >6</TD><TD >7</TD><TD >8</TD><TD >9</TD><TD >10</TD></tr><tr><TD ><font color=blue>5</font></TD><TD >6</TD><TD >7</TD><TD >8</TD><TD >9</TD><TD >10</TD><TD >11</TD></tr><tr><TD ><font color=blue>6</font></TD><TD >7</TD><TD >8</TD><TD >9</TD><TD >10</TD><TD >11</TD><TD >12</TD></tr>\n" );
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document.write( "Eg: adding a blue <font color=blue>1</font> with a red <font color=red>6</font> gets <font color=blue>1</font>+<font color=red>6</font> = 7 (top right corner). \r<BR>\n" );
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document.write( "That dice chart shown above leads to this probability distribution table.<BR>\n" );
document.write( "X = net winnings = prize - cost<BR>\n" );
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document.write( "<tr><TD >X</TD><TD >P(X)</TD></tr><tr><TD >5</TD><TD >2/36</TD></tr><tr><TD >4</TD><TD >4/36</TD></tr><tr><TD >3</TD><TD >6/36</TD></tr><tr><TD >0</TD><TD >8/36</TD></tr><tr><TD >-5</TD><TD >16/36</TD></tr>\n" );
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document.write( "Example: The sum 2 and 12 show up once each, so a total of 1+1 = 2 times. That leads to 2/36 for X = 5 (the amount the player walks away with after winning $10).<BR>\n" );
document.write( "I'll avoid reducing to keep each denominator the same.<BR>\n" );
document.write( "Spreadsheet software is recommended to quickly compute values, and to organize the data in a tidy fashion.\r<BR>\n" );
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document.write( "Form a third column multiplying each X and corresponding P(X) value.<BR>\n" );
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document.write( "<tr><TD >X</TD><TD >P(X)</TD><TD >X*P(X)</TD></tr><tr><TD >5</TD><TD >2/36</TD><TD >10/36</TD></tr><tr><TD >4</TD><TD >4/36</TD><TD >16/36</TD></tr><tr><TD >3</TD><TD >6/36</TD><TD >18/36</TD></tr><tr><TD >0</TD><TD >8/36</TD><TD >0/36</TD></tr><tr><TD >-5</TD><TD >16/36</TD><TD >-80/36</TD></tr>\n" );
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document.write( "Then add up the items in that third column.<BR>\n" );
document.write( "E[X] = expected value = expected net winnings<BR>\n" );
document.write( "E[X] = sum of the X*P(X) values<BR>\n" );
document.write( "E[X] = 10/36 + 16/36 + 18/36 + 0/36 + (-80/36)<BR>\n" );
document.write( "E[X] = (10+16+18+0+(-80))/36<BR>\n" );
document.write( "E[X] = -36/36<BR>\n" );
document.write( "E[X] = -1\r<BR>\n" );
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document.write( "Expected value = -1<BR>\n" );
document.write( "Interpretation: The player expects to lose $1, on average, each time they play the game.\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "The negative expected value is in reference to the player. Meaning the game favors the person hosting the game (and not the person playing the game).\r<BR>\n" );
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document.write( "This is typical in many settings. Casinos make sure that they have the odds tilted in their favor to earn the most money possible. <BR>\n" );
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