<PRE><B>Question 971791</B>
a pair if dice is is. what that probability that the sum is equal to: 

A. 5. 
&lt;pre&gt;Here are all 36 possible ways the pair of dice can land.  The
cases when their sum is equal to 5 are in red:

&lt;b&gt;
(1,1)  (1,2)  (1,3)  &lt;font color=&#39;red&#39;&gt;(1,4)&lt;/font&gt;  (1,5)  (1,6)

(2,1)  (2,2)  &lt;font color=&#39;red&#39;&gt;(2,3)&lt;/font&gt;  (2,4)  (2,5)  (2,6)

(3,1)  &lt;font color=&#39;red&#39;&gt;(3,2)&lt;/font&gt;  (3,3)  (3,4)  (3,5)  (3,6)

&lt;font color=&#39;red&#39;&gt;(4,1)&lt;/font&gt;  (4,2)  (4,3)  (4,4)  (4,5)  (4,6)

(5,1)  (5,2)  (5,3)  (5,4)  (5,5)  (5,6)

(6,1)  (6,2)  (6,3)  (6,4)  (6,5)  (6,6)

&lt;/b&gt;
So the probability that they will land such that their sum is
equal to 5 is those 4 ways out of the 36 possible rolls.
4 out of 36 is a probability of {{{4/36}}} which reduces to {{{1/9}}}.

---------------
&lt;/pre&gt;
B. 10.
&lt;pre&gt;Here again are all 36 possible ways the pair of dice can land.  This
time the cases when their sum is equal to 10 are in red:

&lt;b&gt;
(1,1)  (1,2)  (1,3)  (1,4)  (1,5)  (1,6)

(2,1)  (2,2)  (2,3)  (2,4)  (2,5)  (2,6)

(3,1)  (3,2)  (3,3)  (3,4)  (3,5)  (3,6)

(4,1)  (4,2)  (4,3)  (4,4)  (4,5)  &lt;font color=&#39;red&#39;&gt;(4,6)&lt;/font&gt;

(5,1)  (5,2)  (5,3)  (5,4)  &lt;font color=&#39;red&#39;&gt;(5,5)&lt;/font&gt;  (5,6)

(6,1)  (6,2)  (6,3)  &lt;font color=&#39;red&#39;&gt;(6,4)&lt;/font&gt;  (6,5)  (6,6)

&lt;/b&gt;
So the probability that they will land such that their sum is
equal to 10 is those 3 ways out of the 36 possible rolls.
3 out of 36 is a probability of {{{3/36}}} which reduces to {{{1/12}}}.

-------------------
&lt;/pre&gt;
c. at most 9. 
&lt;pre&gt;Here again are all 36 possible ways the pair of dice can land.  This time the
cases when their sum is at most 9 are in red:

&lt;b&gt;
&lt;font color=&#39;red&#39;&gt;(1,1)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(1,2)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(1,3)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(1,4)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(1,5)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(1,6)&lt;/font&gt;

&lt;font color=&#39;red&#39;&gt;(2,1)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(2,2)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(2,3)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(2,4)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(2,5)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(2,6)&lt;/font&gt;

&lt;font color=&#39;red&#39;&gt;(3,1)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(3,2)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(3,3)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(3,4)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(3,5)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(3,6)&lt;/font&gt;

&lt;font color=&#39;red&#39;&gt;(4,1)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(4,2)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(4,3)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(4,4)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(4,5)&lt;/font&gt;  (4,6)

&lt;font color=&#39;red&#39;&gt;(5,1)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(5,2)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(5,3)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(5,4)&lt;/font&gt;  (5,5)  (5,6)

&lt;font color=&#39;red&#39;&gt;(6,1)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(6,2)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(6,3)&lt;/font&gt;  (6,4)  (6,5)  (6,6)

&lt;/b&gt;
So the probability that they will land such that their sum is
at most 9 is those 30 ways out of the 36 possible rolls.
30 out of 36 is a probability of {{{30/36}}} which reduces to {{{5/6}}}.

------------------------
&lt;/pre&gt;
d. at least 8.
&lt;pre&gt;Here once more are all 36 possible ways the pair of dice can land.  This
time the cases when their sum is at least 8 are in red:

&lt;b&gt;
(1,1)  (1,2)  (1,3)  (1,4)  (1,5)  (1,6)

(2,1)  (2,2)  (2,3)  (2,4)  (2,5)  &lt;font color=&#39;red&#39;&gt;(2,6)&lt;/font&gt;

(3,1)  (3,2)  (3,3)  (3,4)  &lt;font color=&#39;red&#39;&gt;(3,5)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(3,6)&lt;/font&gt;

(4,1)  (4,2)  (4,3)  &lt;font color=&#39;red&#39;&gt;(4,4)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(4,5)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(4,6)&lt;/font&gt;

(5,1)  (5,2)  &lt;font color=&#39;red&#39;&gt;(5,3)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(5,4)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(5,5)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(5,6)&lt;/font&gt;

(6,1)  &lt;font color=&#39;red&#39;&gt;(6,2)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(6,3)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(6,4)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(6,5)&lt;/font&gt;  &lt;font color=&#39;red&#39;&gt;(6,6)&lt;/font&gt;

&lt;/b&gt;
So the probability that they will land such that their sum is
at least 8 is those 15 ways out of the 36 possible rolls.
15 out of 36 is a probability of {{{15/36}}} which reduces to {{{5/12}}}.

Edwin&lt;/pre&gt;</PRE>