Question 1200826
<font color=black size=3>
Part (a)
Refer to this sum of dice chart.
<table border=1 cellpadding=10><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></table>
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). 


That dice chart shown above leads to this probability distribution table.
X = net winnings = prize - cost
<table border = "1" cellpadding = "5"><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></table>
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).
I'll avoid reducing to keep each denominator the same.
Spreadsheet software is recommended to quickly compute values, and to organize the data in a tidy fashion.


Form a third column multiplying each X and corresponding P(X) value.
<table border = "1" cellpadding = "5"><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></table>
Then add up the items in that third column.
E[X] = expected value = expected net winnings
E[X] = sum of the X*P(X) values
E[X] = 10/36 + 16/36 + 18/36 + 0/36 + (-80/36)
E[X] = (10+16+18+0+(-80))/36
E[X] = -36/36
E[X] = -1


Expected value = -1
Interpretation: The player expects to lose $1, on average, each time they play the game.


=============================================================================================================
Part (b)


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).


This is typical in many settings. Casinos make sure that they have the odds tilted in their favor to earn the most money possible. 
</font>