•
The number obtained on the first cast is equally likely to be 1, 2, 3, 4, 5 or 6.
•
The number obtained on the second cast is equally likely to be 1, 2, 3, 4, 5 or 6.
•
The number obtained on the first (second) cast has no influence on the number obtained on
the second (first) cast.
The 36 possible ordered results of the two casts are displayed below, where, for example,
(5
,
3)
means that the first die landed 5 and the second die landed 3. This is different from
(3
,
5)
.
Number from
Number from second cast
first cast
1
2
3
4
5
6
1
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(1,6)
2
(2,1)
(2,2)
(2,3)
(2,4)
(2,5)
(2,6)
3
(3,1)
(3,2)
(3,3)
(3,4)
(3,5)
(3,6)
4
(4,1)
(4,2)
(4,3)
(4,4)
(4,5)
(4,6)
5
(5,1)
(5,2)
(5,3)
(5,4)
(5,5)
(5,6)
6
(6,1)
(6,2)
(6,3)
(6,4)
(6,5)
(6,6)
Just like in the blood type example, b/c of my assumptions, I conclude that these 36 possibilities
are equally likely. We will do a number of calculations now.
For ease of presentation, define
X
1
to be the number obtained on the first cast of the die and let
X
2
denote the number obtained on the second cast of the die.
We call
X
1
and
X
2
random variables
, which means that to each possible outcome of the CM
they assign a number. Every random variable has a
probability distribution
which is simply a
listing of its possible values along with the probability of each value. Note that
X
1
and
X
2
have
the same probability distribution; a fact we describe by saying that they are
identically distributed
,