Question 1078018
Two dice are rolled once. What is the probability that the 
two numbers that show...

<pre><font size = 6 color = "indigo">
<b>
Here are all the possible rolls with a 
red die and a green die:
</b>
(<font color="red">1</font>,<font color="green">1</font>) (<font color="red">1</font>,<font color="green">2</font>) (<font color="red">1</font>,<font color="green">3</font>) (<font color="red">1</font>,<font color="green">4</font>) (<font color="red">1</font>,<font color="green">5</font>) (<font color="red">1</font>,<font color="green">6</font>)
 
(<font color="red">2</font>,<font color="green">1</font>) (<font color="red">2</font>,<font color="green">2</font>) (<font color="red">2</font>,<font color="green">3</font>) (<font color="red">2</font>,<font color="green">4</font>) (<font color="red">2</font>,<font color="green">5</font>) (<font color="red">2</font>,<font color="green">6</font>) 

(<font color="red">3</font>,<font color="green">1</font>) (<font color="red">3</font>,<font color="green">2</font>) (<font color="red">3</font>,<font color="green">3</font>) (<font color="red">3</font>,<font color="green">4</font>) (<font color="red">3</font>,<font color="green">5</font>) (<font color="red">3</font>,<font color="green">6</font>)
 
(<font color="red">4</font>,<font color="green">1</font>) (<font color="red">4</font>,<font color="green">2</font>) (<font color="red">4</font>,<font color="green">3</font>) (<font color="red">4</font>,<font color="green">4</font>) (<font color="red">4</font>,<font color="green">5</font>) (<font color="red">4</font>,<font color="green">6</font>)
 
(<font color="red">5</font>,<font color="green">1</font>) (<font color="red">5</font>,<font color="green">2</font>) (<font color="red">5</font>,<font color="green">3</font>) (<font color="red">5</font>,<font color="green">4</font>) (<font color="red">5</font>,<font color="green">5</font>) (<font color="red">5</font>,<font color="green">6</font>)
 
(<font color="red">6</font>,<font color="green">1</font>) (<font color="red">6</font>,<font color="green">2</font>) (<font color="red">6</font>,<font color="green">3</font>) (<font color="red">6</font>,<font color="green">4</font>) (<font color="red">6</font>,<font color="green">5</font>) (<font color="red">6</font>,<font color="green">6</font>) 
<b>
</pre>
a) have a sum of 7?
<pre>
There are 36 possible rolls.  Now I 
will underline just those which have
a sum of 7:
</b>
(<font color="red">1</font>,<font color="green">1</font>) (<font color="red">1</font>,<font color="green">2</font>) (<font color="red">1</font>,<font color="green">3</font>) (<font color="red">1</font>,<font color="green">4</font>) (<font color="red">1</font>,<font color="green">5</font>) <u><b>(<font color="red">1</font>,<font color="green">6</font>)</u></b>
 
(<font color="red">2</font>,<font color="green">1</font>) (<font color="red">2</font>,<font color="green">2</font>) (<font color="red">2</font>,<font color="green">3</font>) (<font color="red">2</font>,<font color="green">4</font>) <u><b>(<font color="red">2</font>,<font color="green">5</font>)</u></b> (<font color="red">2</font>,<font color="green">6</font>) 

(<font color="red">3</font>,<font color="green">1</font>) (<font color="red">3</font>,<font color="green">2</font>) (<font color="red">3</font>,<font color="green">3</font>) <u><b>(<font color="red">3</font>,<font color="green">4</font>)</u></b> (<font color="red">3</font>,<font color="green">5</font>) (<font color="red">3</font>,<font color="green">6</font>)
 
(<font color="red">4</font>,<font color="green">1</font>) (<font color="red">4</font>,<font color="green">2</font>) <u><b>(<font color="red">4</font>,<font color="green">3</font>)</u></b> (<font color="red">4</font>,<font color="green">4</font>) (<font color="red">4</font>,<font color="green">5</font>) (<font color="red">4</font>,<font color="green">6</font>)
 
(<font color="red">5</font>,<font color="green">1</font>) <u><b>(<font color="red">5</font>,<font color="green">2</font>)</u></b> (<font color="red">5</font>,<font color="green">3</font>) (<font color="red">5</font>,<font color="green">4</font>) (<font color="red">5</font>,<font color="green">5</font>) (<font color="red">5</font>,<font color="green">6</font>)
 
<u><b>(<font color="red">6</font>,<font color="green">1</font>)</u></b> (<font color="red">6</font>,<font color="green">2</font>) (<font color="red">6</font>,<font color="green">3</font>) (<font color="red">6</font>,<font color="green">4</font>) (<font color="red">6</font>,<font color="green">5</font>) (<font color="red">6</font>,<font color="green">6</font>)
<b>
There are 36 possible rolls.  So 
there are 6 possible rolls out of
the 36 which have that property.

Therefore the probability is {{{6/36}}} 
which reduces to {{{1/6}}}
</pre>
b) have a sum of at most 5?<pre>
There are 36 possible rolls.  Now I 
will underline just those which have
a sum of at most 5:
<pre>
<u><b>(<font color="red">1</font>,<font color="green">1</font>) (<font color="red">1</font>,<font color="green">2</font>) (<font color="red">1</font>,<font color="green">3</font>) (<font color="red">1</font>,<font color="green">4</font>)</u></b> (<font color="red">1</font>,<font color="green">5</font>) (<font color="red">1</font>,<font color="green">6</font>)
 
<u><b>(<font color="red">2</font>,<font color="green">1</font>) (<font color="red">2</font>,<font color="green">2</font>) (<font color="red">2</font>,<font color="green">3</font>)</u></b> (<font color="red">2</font>,<font color="green">4</font>) (<font color="red">2</font>,<font color="green">5</font>) (<font color="red">2</font>,<font color="green">6</font>) 

<u><b>(<font color="red">3</font>,<font color="green">1</font>) (<font color="red">3</font>,<font color="green">2</font>)</u></b> (<font color="red">3</font>,<font color="green">3</font>) (<font color="red">3</font>,<font color="green">4</font>) (<font color="red">3</font>,<font color="green">5</font>) (<font color="red">3</font>,<font color="green">6</font>)
 
<u><b>(<font color="red">4</font>,<font color="green">1</font>)</u></b> (<font color="red">4</font>,<font color="green">2</font>) (<font color="red">4</font>,<font color="green">3</font>) (<font color="red">4</font>,<font color="green">4</font>) (<font color="red">4</font>,<font color="green">5</font>) (<font color="red">4</font>,<font color="green">6</font>)
 
(<font color="red">5</font>,<font color="green">1</font>) (<font color="red">5</font>,<font color="green">2</font>) (<font color="red">5</font>,<font color="green">3</font>) (<font color="red">5</font>,<font color="green">4</font>) (<font color="red">5</font>,<font color="green">5</font>) (<font color="red">5</font>,<font color="green">6</font>)
 
(<font color="red">6</font>,<font color="green">1</font>) (<font color="red">6</font>,<font color="green">2</font>) (<font color="red">6</font>,<font color="green">3</font>) (<font color="red">6</font>,<font color="green">4</font>) (<font color="red">6</font>,<font color="green">5</font>) (<font color="red">6</font>,<font color="green">6</font>) 
<b>
There are 36 possible rolls.  So 
there are 10 possible rolls out of
the 36 which have that property.

Therefore the probability is {{{10/36}}} 
which reduces to {{{5/18}}}
</pre></pre>
c) have a sum which is even and at least 8?
<pre>
Now I will underline just those which have
a sum which is even and at least 8:
</b>
(<font color="red">1</font>,<font color="green">1</font>) (<font color="red">1</font>,<font color="green">2</font>) (<font color="red">1</font>,<font color="green">3</font>) (<font color="red">1</font>,<font color="green">4</font>) (<font color="red">1</font>,<font color="green">5</font>) (<font color="red">1</font>,<font color="green">6</font>)
 
(<font color="red">2</font>,<font color="green">1</font>) (<font color="red">2</font>,<font color="green">2</font>) (<font color="red">2</font>,<font color="green">3</font>) (<font color="red">2</font>,<font color="green">4</font>) (<font color="red">2</font>,<font color="green">5</font>) <u><b>(<font color="red">2</font>,<font color="green">6</font>)</u></b> 

(<font color="red">3</font>,<font color="green">1</font>) (<font color="red">3</font>,<font color="green">2</font>) (<font color="red">3</font>,<font color="green">3</font>) (<font color="red">3</font>,<font color="green">4</font>) <u><b>(<font color="red">3</font>,<font color="green">5</font>)</u></b> (<font color="red">3</font>,<font color="green">6</font>)
 
(<font color="red">4</font>,<font color="green">1</font>) (<font color="red">4</font>,<font color="green">2</font>) (<font color="red">4</font>,<font color="green">3</font>) <u><b>(<font color="red">4</font>,<font color="green">4</font>)</u></b> (<font color="red">4</font>,<font color="green">5</font>) <u><b>(<font color="red">4</font>,<font color="green">6</font>)</u></b>
 
(<font color="red">5</font>,<font color="green">1</font>) (<font color="red">5</font>,<font color="green">2</font>) <u><b>(<font color="red">5</font>,<font color="green">3</font>)</u></b> (<font color="red">5</font>,<font color="green">4</font>) <u><b>(<font color="red">5</font>,<font color="green">5</font>)</u></b> (<font color="red">5</font>,<font color="green">6</font>)
 
(<font color="red">6</font>,<font color="green">1</font>) <u><b>(<font color="red">6</font>,<font color="green">2</font>)</u></b> (<font color="red">6</font>,<font color="green">3</font>) <u><b>(<font color="red">6</font>,<font color="green">4</font>)</u></b> (<font color="red">6</font>,<font color="green">5</font>) <u><b>(<font color="red">6</font>,<font color="green">6</font>)</u></b> 
<b>
There are 36 possible rolls.  So 
there are 9 possible rolls out of
the 36 which have both properties.

Therefore the probability is {{{9/36}}} 
which reduces to {{{1/4}}}

Edwin</pre></b></font>