Question 1149654
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The expected value E(X) is equal to the sum of all the X*P(X) values. 


In math notation,
*[Tex \Large \displaystyle E(X) = \sum X*P(X)]

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The probability of landing on heads is 1/2
The probability of rolling any given side of a die is 1/6
The probability of getting heads and getting any given side of a die is (1/2)*(1/6) = 1/12


If the coin lands on heads
<table border = "1" cellpadding="10">
<tr><td>X</td><td>P(X)</td><td>X*P(X)</td></tr>
<tr><td>1</td><td>1/12</td><td>1*(1/12) = 1/12</td></tr>
<tr><td>2</td><td>1/12</td><td>2*(1/12) = 2/12</td></tr>
<tr><td>3</td><td>1/12</td><td>3*(1/12) = 3/12</td></tr>
<tr><td>4</td><td>1/12</td><td>4*(1/12) = 4/12</td></tr>
<tr><td>5</td><td>1/12</td><td>5*(1/12) = 5/12</td></tr>
<tr><td>6</td><td>1/12</td><td>6*(1/12) = 6/12</td></tr>
</table>
Sum the values in the third column
1/12 + 2/12 + 3/12 + 4/12 + 5/12 + 6/12 = (1+2+3+4+5+6)/12 = 21/12 = 7/4 = <font color=blue>1.75</font>
We'll use the value <font color=blue>1.75</font> later on.


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The probability of landing on tails is 1/2


The chart below represents rolling two dice, one red and one blue, then summing the values. Each sum is shown in black. 
<img src = "https://i.imgur.com/KhIVE5Z.png">
Use the chart to see that
probability of rolling a 2 is 1/36
probability of rolling a 3 is 2/36
probability of rolling a 4 is 3/36
probability of rolling a 5 is 4/36
probability of rolling a 6 is 5/36
probability of rolling a 7 is 6/36
probability of rolling a 8 is 5/36
probability of rolling a 9 is 4/36
probability of rolling a 10 is 3/36
probability of rolling a 11 is 2/36
probability of rolling a 12 is 1/36


which helps us construct the table below
<table border = "1" cellpadding="10">
<tr><td>X</td><td>P(X)</td><td>X*P(X)</td></tr>
<tr><td>2</td><td>(1/2)*(1/36)=1/72</td><td>(2)*(1/72) = 2/72</td></tr>
<tr><td>3</td><td>(1/2)*(2/36)=2/72</td><td>(3)*(2/72) = 6/72</td></tr>
<tr><td>4</td><td>(1/2)*(3/36)=3/72</td><td>(4)*(3/72) = 12/72</td></tr>
<tr><td>5</td><td>(1/2)*(4/36)=4/72</td><td>(5)*(4/72) = 20/72</td></tr>
<tr><td>6</td><td>(1/2)*(5/36)=5/72</td><td>(6)*(5/72) = 30/72</td></tr>
<tr><td>7</td><td>(1/2)*(6/36)=6/72</td><td>(7)*(6/72) = 42/72</td></tr>
<tr><td>8</td><td>(1/2)*(5/36)=5/72</td><td>(8)*(5/72) = 40/72</td></tr>
<tr><td>9</td><td>(1/2)*(4/36)=4/72</td><td>(9)*(4/72) = 36/72</td></tr>
<tr><td>10</td><td>(1/2)*(3/36)=3/72</td><td>(10)*(3/72) = 30/72</td></tr>
<tr><td>11</td><td>(1/2)*(2/36)=2/72</td><td>(11)*(2/72) = 22/72</td></tr>
<tr><td>12</td><td>(1/2)*(1/36)=1/72</td><td>(12)*(1/72) = 12/72</td></tr>
</table>
Sum the values in the third column
2/72 + 6/72 + 12/72 + 20/72 + 30/72 + 42/72 + 40/72 + 36/72 + 30/72 + 22/72 + 12/72 = (2+6+12+20+30+42+40+36+30+22+12)/72 = 252/72 = 7/2 = <font color=green>3.5</font>
We'll use the value <font color=green>3.5</font> later on.


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Add the two sub-results from each section
<font color=blue>1.75</font> + <font color=green>3.5</font> = <font color=red>5.25</font>



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<font color=red size=4>Answer: 5.25</font>
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