Question 1197919
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Part (a)


<table border = "1" cellpadding = "5"><tr><td>Spinning a...</td><td>Points Won</td><td>Probability</td></tr><tr><td>1</td><td>1</td><td>1/12</td></tr><tr><td>2</td><td>-2</td><td>1/12</td></tr><tr><td>3</td><td>3</td><td>1/12</td></tr><tr><td>4</td><td>-4</td><td>1/12</td></tr><tr><td>5</td><td>5</td><td>1/12</td></tr><tr><td>6</td><td>-6</td><td>1/12</td></tr><tr><td>7</td><td>7</td><td>1/12</td></tr><tr><td>8</td><td>-8</td><td>1/12</td></tr><tr><td>9</td><td>9</td><td>1/12</td></tr><tr><td>10</td><td>-10</td><td>1/12</td></tr><tr><td>11</td><td>11</td><td>1/12</td></tr><tr><td>12</td><td>-12</td><td>1/12</td></tr></table>


X = number of points won
P(X) = probability of winning X points<table border = "1" cellpadding = "5"><tr><td>X</td><td>P(X)</td></tr><tr><td>1</td><td>1/12</td></tr><tr><td>-2</td><td>1/12</td></tr><tr><td>3</td><td>1/12</td></tr><tr><td>-4</td><td>1/12</td></tr><tr><td>5</td><td>1/12</td></tr><tr><td>-6</td><td>1/12</td></tr><tr><td>7</td><td>1/12</td></tr><tr><td>-8</td><td>1/12</td></tr><tr><td>9</td><td>1/12</td></tr><tr><td>-10</td><td>1/12</td></tr><tr><td>11</td><td>1/12</td></tr><tr><td>-12</td><td>1/12</td></tr></table>


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Part (b)


Form a third column labeled X*P(X)
As you can probably guess, this column represents us multiplying each X and P(X) value
Eg: -2*(1/12) = -2/12
I'll leave the fractions un-reduced<table border = "1" cellpadding = "5"><tr><td>X</td><td>P(X)</td><td>X*P(X)</td></tr><tr><td>1</td><td>1/12</td><td>1/12</td></tr><tr><td>-2</td><td>1/12</td><td>-2/12</td></tr><tr><td>3</td><td>1/12</td><td>3/12</td></tr><tr><td>-4</td><td>1/12</td><td>-4/12</td></tr><tr><td>5</td><td>1/12</td><td>5/12</td></tr><tr><td>-6</td><td>1/12</td><td>-6/12</td></tr><tr><td>7</td><td>1/12</td><td>7/12</td></tr><tr><td>-8</td><td>1/12</td><td>-8/12</td></tr><tr><td>9</td><td>1/12</td><td>9/12</td></tr><tr><td>-10</td><td>1/12</td><td>-10/12</td></tr><tr><td>11</td><td>1/12</td><td>11/12</td></tr><tr><td>-12</td><td>1/12</td><td>-12/12</td></tr></table>


Then we add up everything in the X*P(X) column.
We'll add the numerators


1+(-2)+3+(-4)+5+(-6)+7+(-8)+9+(-10)+11+(-12) = -6 which is the numerator in the answer before reducing.


Adding up all the fractions in the X*P(X) column leads to -6/12 = -1/2
The player expects to lose, on average, 1/2 = 0.5 points per game.


Answer: -1/2 = -0.5


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Part (c)


The game is not fair because the expected value (in part b) is not zero.
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