Question 487189
<pre><b>
You must make a chart for the sample space of all possible
dice throws:
 
Here is the sample space. It contains 64 possible outcomes:

(1,1)  (1,2)   (1,3)  (1,4)   (1,5)  (1,6)  (1,7)  (1,8)

(2,1)  (2,2)   (2,3)  (2,4)   (2,5)  (2,6)  (2,7)  (2,8)

(3,1)  (3,2)   (3,3)  (3,4)   (3,5)  (3,6)  (3,7)  (3,8)

(4,1)  (4,2)   (4,3)  (4,4)   (4,5)  (4,6)  (4,7)  (4,8)

(5,1)  (5,2)   (5,3)  (5,4)   (5,5)  (5,6)  (5,7)  (5,8)

(6,1)  (6,2)   (6,3)  (6,4)   (6,5)  (6,6)  (6,7)  (6,8)

(7,1)  (7,2)   (7,3)  (7,4)   (7,5)  (7,6)  (7,7)  (7,8)

(8,1)  (8,2)   (8,3)  (8,4)   (8,5)  (8,6)  (8,7)  (8,8)

You'll find that all throws with a given sum are on a diagonal.

-----------------------------------
a:P(4)
The throws with sum 4 are in red:

(1,1)  (1,2)   <font color="red">(1,3)</font>  (1,4)   (1,5)  (1,6)  (1,7)  (1,8)

(2,1)  <font color="red">(2,2)</font>   (2,3)  (2,4)   (2,5)  (2,6)  (2,7)  (2,8)

<font color="red">(3,1)</font>  (3,2)   (3,3)  (3,4)   (3,5)  (3,6)  (3,7)  (3,8)

(4,1)  (4,2)   (4,3)  (4,4)   (4,5)  (4,6)  (4,7)  (4,8)

(5,1)  (5,2)   (5,3)  (5,4)   (5,5)  (5,6)  (5,7)  (5,8)

(6,1)  (6,2)   (6,3)  (6,4)   (6,5)  (6,6)  (6,7)  (6,8)

(7,1)  (7,2)   (7,3)  (7,4)   (7,5)  (7,6)  (7,7)  (7,8)

(8,1)  (8,2)   (8,3)  (8,4)   (8,5)  (8,6)  (8,7)  (8,8)

Therefore P(4) is 3 out of 64 or {{{3/64}}}

-------------------------------------------------
b:P(6)
The throws with sum 6 are in red:


(1,1)  (1,2)   (1,3)  (1,4)   <font color="red">(1,5)</font>  (1,6)  (1,7)  (1,8)

(2,1)  (2,2)   (2,3)  <font color="red">(2,4)</font>   (2,5)  (2,6)  (2,7)  (2,8)

(3,1)  (3,2)   <font color="red">(3,3)</font>  (3,4)   (3,5)  (3,6)  (3,7)  (3,8)

(4,1)  <font color="red">(4,2)</font>   (4,3)  (4,4)   (4,5)  (4,6)  (4,7)  (4,8)

<font color="red">(5,1)</font>  (5,2)   (5,3)  (5,4)   (5,5)  (5,6)  (5,7)  (5,8)

(6,1)  (6,2)   (6,3)  (6,4)   (6,5)  (6,6)  (6,7)  (6,8)

(7,1)  (7,2)   (7,3)  (7,4)   (7,5)  (7,6)  (7,7)  (7,8)

(8,1)  (8,2)   (8,3)  (8,4)   (8,5)  (8,6)  (8,7)  (8,8)

Therefore P(6) is 5 out of 64 or {{{5/64}}}

-------------------------------------------------

The throws with sum 10 are in red:

(1,1)  (1,2)   (1,3)  (1,4)   (1,5)  (1,6)  (1,7)  (1,8)

(2,1)  (2,2)   (2,3)  (2,4)   (2,5)  (2,6)  (2,7)  <font color="red">(2,8)</font>

(3,1)  (3,2)   (3,3)  (3,4)   (3,5)  (3,6)  <font color="red">(3,7)</font>  (3,8)

(4,1)  (4,2)   (4,3)  (4,4)   (4,5)  <font color="red">(4,6)</font>  (4,7)  (4,8)

(5,1)  (5,2)   (5,3)  (5,4)   <font color="red">(5,5)</font>  (5,6)  (5,7)  (5,8)

(6,1)  (6,2)   (6,3)  <font color="red">(6,4)</font>   (6,5)  (6,6)  (6,7)  (6,8)

(7,1)  (7,2)   <font color="red">(7,3)</font>  (7,4)   (7,5)  (7,6)  (7,7)  (7,8)

(8,1)  <font color="red">(8,2)</font>   (8,3)  (8,4)   (8,5)  (8,6)  (8,7)  (8,8)

Therefore P(10) is 7 out of 64 or {{{7/64}}}

-------------------------------------------------

The throws with sum 11 are in red:

(1,1)  (1,2)   (1,3)  (1,4)   (1,5)  (1,6)  (1,7)  (1,8)

(2,1)  (2,2)   (2,3)  (2,4)   (2,5)  (2,6)  (2,7)  (2,8)

(3,1)  (3,2)   (3,3)  (3,4)   (3,5)  (3,6)  (3,7)  <font color="red">(3,8)</font>

(4,1)  (4,2)   (4,3)  (4,4)   (4,5)  (4,6)  <font color="red">(4,7)</font>  (4,8)

(5,1)  (5,2)   (5,3)  (5,4)   (5,5)  <font color="red">(5,6)</font>  (5,7)  (5,8)

(6,1)  (6,2)   (6,3)  (6,4)   <font color="red">(6,5)</font>  (6,6)  (6,7)  (6,8)

(7,1)  (7,2)   (7,3)  <font color="red">(7,4)</font>   (7,5)  (7,6)  (7,7)  (7,8)

(8,1)  (8,2)   <font color="red">(8,3)</font>  (8,4)   (8,5)  (8,6)  (8,7)  (8,8)

Therefore P(11) is 6 out of 64 or {{{6/64}}} which reduces to {{{3/32}}}

-------------------------------------------------

The throws with sum 14 are in red:

(1,1)  (1,2)   (1,3)  (1,4)   (1,5)  (1,6)  (1,7)  (1,8)

(2,1)  (2,2)   (2,3)  (2,4)   (2,5)  (2,6)  (2,7)  (2,8)

(3,1)  (3,2)   (3,3)  (3,4)   (3,5)  (3,6)  (3,7)  (3,8)

(4,1)  (4,2)   (4,3)  (4,4)   (4,5)  (4,6)  (4,7)  (4,8)

(5,1)  (5,2)   (5,3)  (5,4)   (5,5)  (5,6)  (5,7)  (5,8)

(6,1)  (6,2)   (6,3)  (6,4)   (6,5)  (6,6)  (6,7)  <font color="red">(6,8)</font>

(7,1)  (7,2)   (7,3)  (7,4)   (7,5)  (7,6)  <font color="red">(7,7)</font>  (7,8)

(8,1)  (8,2)   (8,3)  (8,4)   (8,5)  <font color="red">(8,6)</font>  (8,7)  (8,8)

Therefore P(14) is 3 out of 64 or {{{3/64}}}

-------------------------------------------------

The throw with sum 16 is in red:

(1,1)  (1,2)   (1,3)  (1,4)   (1,5)  (1,6)  (1,7)  (1,8)

(2,1)  (2,2)   (2,3)  (2,4)   (2,5)  (2,6)  (2,7)  (2,8)

(3,1)  (3,2)   (3,3)  (3,4)   (3,5)  (3,6)  (3,7)  (3,8)

(4,1)  (4,2)   (4,3)  (4,4)   (4,5)  (4,6)  (4,7)  (4,8)

(5,1)  (5,2)   (5,3)  (5,4)   (5,5)  (5,6)  (5,7)  (5,8)

(6,1)  (6,2)   (6,3)  (6,4)   (6,5)  (6,6)  (6,7)  (6,8)

(7,1)  (7,2)   (7,3)  (7,4)   (7,5)  (7,6)  (7,7)  (7,8)

(8,1)  (8,2)   (8,3)  (8,4)   (8,5)  (8,6)  (8,7)  <font color="red">(8,8)</font>

Therefore P(16) is 1 out of 64 or {{{1/64}}}

-------------------------------------------------
Edwin</pre>