Question 1197519
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Question 1, Part (a)


This is one way to write out the table of possible outcomes.
<table border = "1" cellpadding = "5"><tr><td></td><td><font color=blue>1</font></td><td><font color=blue>2</font></td><td><font color=blue>3</font></td><td><font color=blue>4</font></td><td><font color=blue>5</font></td><td><font color=blue>6</font></td></tr><tr><td><font color=red>1</font></td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td></tr><tr><td><font color=red>2</font></td><td>2</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td></tr><tr><td><font color=red>3</font></td><td>3</td><td>3</td><td>3</td><td>4</td><td>5</td><td>6</td></tr><tr><td><font color=red>4</font></td><td>4</td><td>4</td><td>4</td><td>4</td><td>5</td><td>6</td></tr><tr><td><font color=red>5</font></td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>6</td></tr><tr><td><font color=red>6</font></td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td></tr></table>Example: We roll a <font color=blue>6 on the blue die</font> and a <font color=red>1 on the red die</font>. The result is 6 since it's the larger of the two outcomes (top right corner of the table).


The possible outcomes are: 1,2,3,4,5,6
Let X be those possible outcomes.


Here's the frequency chart
<table border = "1" cellpadding = "5"><tr><td>X</td><td>Frequency</td></tr><tr><td>1</td><td>1</td></tr><tr><td>2</td><td>3</td></tr><tr><td>3</td><td>5</td></tr><tr><td>4</td><td>7</td></tr><tr><td>5</td><td>9</td></tr><tr><td>6</td><td>11</td></tr></table>
We'll divide each frequency over 36 since there are 6*6 = 36 outcomes.


Therefore, we have this probability mass function (PMF).
<table border = "1" cellpadding = "5"><tr><td>X</td><td>P(X)</td></tr><tr><td>1</td><td>1/36</td></tr><tr><td>2</td><td>3/36</td></tr><tr><td>3</td><td>5/36</td></tr><tr><td>4</td><td>7/36</td></tr><tr><td>5</td><td>9/36</td></tr><tr><td>6</td><td>11/36</td></tr></table>I decided not to reduce the fractions so that each could keep the consistent denominator of 36. 


If you want to reduce the fractions, then,
3/36 = 1/12
9/36 = 1/4


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Question 1, Part (b)


Right now I'm blanking on how to do this, so I'll come back to this later. Or I'll let another tutor step in. Sorry for the trouble.


The slight good news is that I managed to find the PMF tables for k = 3 through k = 5 using computer software. But I wasn't able to find a generalized case for any positive integer k value.


PMF for k = 3
<table border = "1" cellpadding = "5"><tr><td>X</td><td>P(X)</td></tr><tr><td>1</td><td>1/216</td></tr><tr><td>2</td><td>7/216</td></tr><tr><td>3</td><td>19/216</td></tr><tr><td>4</td><td>37/216</td></tr><tr><td>5</td><td>61/216</td></tr><tr><td>6</td><td>91/216</td></tr></table>


PMF for k = 4
<table border = "1" cellpadding = "5"><tr><td>X</td><td>P(X)</td></tr><tr><td>1</td><td>1/1296</td></tr><tr><td>2</td><td>15/1296</td></tr><tr><td>3</td><td>65/1296</td></tr><tr><td>4</td><td>175/1296</td></tr><tr><td>5</td><td>369/1296</td></tr><tr><td>6</td><td>671/1296</td></tr></table>


PMF for k = 5
<table border = "1" cellpadding = "5"><tr><td>X</td><td>P(X)</td></tr><tr><td>1</td><td>1/7776</td></tr><tr><td>2</td><td>31/7776</td></tr><tr><td>3</td><td>211/7776</td></tr><tr><td>4</td><td>781/7776</td></tr><tr><td>5</td><td>2101/7776</td></tr><tr><td>6</td><td>4651/7776</td></tr></table>


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Question 2


We'll use the template from the table in question 1, part (a).


Instead of white, I'll use blue. 
We subtract the values in the format B - R
B = blue
R = red
<table border = "1" cellpadding = "5"><tr><td></td><td><font color=blue>1</font></td><td><font color=blue>2</font></td><td><font color=blue>3</font></td><td><font color=blue>4</font></td><td><font color=blue>5</font></td><td><font color=blue>6</font></td></tr><tr><td><font color=red>1</font></td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td><font color=red>2</font></td><td>-1</td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td><font color=red>3</font></td><td>-2</td><td>-1</td><td>0</td><td>1</td><td>2</td><td>3</td></tr><tr><td><font color=red>4</font></td><td>-3</td><td>-2</td><td>-1</td><td>0</td><td>1</td><td>2</td></tr><tr><td><font color=red>5</font></td><td>-4</td><td>-3</td><td>-2</td><td>-1</td><td>0</td><td>1</td></tr><tr><td><font color=red>6</font></td><td>-5</td><td>-4</td><td>-3</td><td>-2</td><td>-1</td><td>0</td></tr></table>
For example, if we roll a <font color=blue>6 on the blue die</font> and a <font color=red>1 on the red die</font>, then <font color=blue>B</font>-<font color=red>R</font> = <font color=blue>6</font>-<font color=red>1</font> = 5 which is in the top right corner of the table.


The outcomes range from -5 to +5 inclusive of the endpoints.
Let X be the result of each difference


<table border = "1" cellpadding = "5"><tr><td>X</td><td>Frequency</td></tr><tr><td>-5</td><td>1</td></tr><tr><td>-4</td><td>2</td></tr><tr><td>-3</td><td>3</td></tr><tr><td>-2</td><td>4</td></tr><tr><td>-1</td><td>5</td></tr><tr><td>0</td><td>6</td></tr><tr><td>1</td><td>5</td></tr><tr><td>2</td><td>4</td></tr><tr><td>3</td><td>3</td></tr><tr><td>4</td><td>2</td></tr><tr><td>5</td><td>1</td></tr></table>
Then we divide each frequency over 36 to form the PMF.
<table border = "1" cellpadding = "5"><tr><td>X</td><td>P(X)</td></tr><tr><td>-5</td><td>1/36</td></tr><tr><td>-4</td><td>2/36</td></tr><tr><td>-3</td><td>3/36</td></tr><tr><td>-2</td><td>4/36</td></tr><tr><td>-1</td><td>5/36</td></tr><tr><td>0</td><td>6/36</td></tr><tr><td>1</td><td>5/36</td></tr><tr><td>2</td><td>4/36</td></tr><tr><td>3</td><td>3/36</td></tr><tr><td>4</td><td>2/36</td></tr><tr><td>5</td><td>1/36</td></tr></table>
Once again, I chose not to reduce the fractions to keep the same denominator (36).
If you want to reduce the fractions, then,
2/36 = 1/18
3/36 = 1/12
4/36 = 1/9
6/36 = 1/6
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