<PRE><B>Question 1064139</B>
Two dice are rolled. What is the probability that:
&lt;pre&gt;
There are 36 possible rolls. I&#39;ll color the &quot;successful&quot; \
rolls red in each case.
&lt;/pre&gt;
a. the numbers that come up have a sum of at least 3?
&lt;pre&gt;
(1,1) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(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) (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;/font&gt;&lt;/b&gt;  

Count the red ones. That&#39;s 35 out of 36 for a 
probability of 35/36.
&lt;/pre&gt;
b. the numbers that come up have a sum of 4 or 10?
&lt;pre&gt;
(1,1) (1,2) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(1,3)&lt;/font&gt;&lt;/b&gt; (1,4) (1,5) (1,6)

(2,1) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(2,2)&lt;/font&gt;&lt;/b&gt; (2,3) (2,4) (2,5) (2,6) 

&lt;font color=&quot;red&quot;&gt;&lt;b&gt;(3,1)&lt;/font&gt;&lt;/b&gt; (3,2) (3,3) (3,4) (3,5) (3,6) 

(4,1) (4,2) (4,3) (4,4) (4,5) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(4,6)&lt;/font&gt;&lt;/b&gt;  

(5,1) (5,2) (5,3) (5,4) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(5,5)&lt;/font&gt;&lt;/b&gt; (5,6) 

(6,1) (6,2) (6,3) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(6,4)&lt;/font&gt;&lt;/b&gt; (6,5) (6,6)  

Count the red ones. That&#39;s 6 out of 36 for a 
probability of 6/36, which reduces to 1/6.
&lt;/pre&gt;
c. the numbers that come up are both odd or have a sum of at least 8?
&lt;pre&gt;
&lt;font color=&quot;red&quot;&gt;&lt;b&gt;(1,1)&lt;/font&gt;&lt;/b&gt; (1,2) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(1,3)&lt;/font&gt;&lt;/b&gt; (1,4) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(1,5)&lt;/font&gt;&lt;/b&gt;  (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(2,6)&lt;/font&gt;&lt;/b&gt;  

&lt;font color=&quot;red&quot;&gt;&lt;b&gt;(3,1)&lt;/font&gt;&lt;/b&gt; (3,2) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(3,3)&lt;/font&gt;&lt;/b&gt; (3,4) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(3,5) (3,6)&lt;/font&gt;&lt;/b&gt; 

(4,1) (4,2) (4,3) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(4,4) (4,5) (4,6)&lt;/b&gt;&lt;/font&gt; 

&lt;font color=&quot;red&quot;&gt;&lt;b&gt;(5,1)&lt;/font&gt;&lt;/b&gt; (5,2) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(5,3) (5,4) (5,5) (5,6)&lt;/font&gt;&lt;/b&gt; 

(6,1) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(6,2) (6,3) (6,4) (6,5) (6,6)&lt;/font&gt;&lt;/b&gt; 

Count the red ones. That&#39;s 21 out of 36 for a 
probability of 21/36, which reduces to 7/12.
&lt;/pre&gt;
d. one number is odd and the other is even or the numbers have a sum of at least 10?
&lt;pre&gt;
(1,1) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(1,2)&lt;/font&gt;&lt;/b&gt; (1,3) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(1,4)&lt;/font&gt;&lt;/b&gt; (1,5) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(1,6)&lt;/font&gt;&lt;/b&gt; 

&lt;font color=&quot;red&quot;&gt;&lt;b&gt;(2,1)&lt;/font&gt;&lt;/b&gt; (2,2) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(2,3)&lt;/font&gt;&lt;/b&gt; (2,4) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(2,5)&lt;/font&gt;&lt;/b&gt; (2,6) 

(3,1) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(3,2)&lt;/font&gt;&lt;/b&gt; (3,3) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(3,4)&lt;/font&gt;&lt;/b&gt; (3,5) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(3,6)&lt;/b&gt;&lt;/font&gt; 

&lt;font color=&quot;red&quot;&gt;&lt;b&gt;(4,1)&lt;/font&gt;&lt;/b&gt; (4,2) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(4,3)&lt;/font&gt;&lt;/b&gt; (4,4) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(4,5)&lt;/font&gt;&lt;/b&gt; &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(4,6)&lt;/font&gt;&lt;/b&gt; 

(5,1) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(5,2)&lt;/font&gt;&lt;/b&gt; (5,3) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(5,4) (5,5) (5,6) 

(6,1)&lt;/font&gt;&lt;/b&gt; (6,2) &lt;font color=&quot;red&quot;&gt;&lt;b&gt;(6,3) (6,4) (6,5) (6,6)&lt;/font&gt;&lt;/b&gt;

Count the red ones. That&#39;s 22 out of 36 for a 
probability of 11/18.

Edwin&lt;/pre&gt;</PRE>