What is the expected return
when we throw two dice? Suppose the number
you throw represents the return, and suppose you
throw the dice thousands of times. What would
you expect your return to be on average?
We can compute the expected return for each possible
result and then sum these results.
Return 
Probability 
Weighted expected
return 
$2 
1/36 = 2.8% 
$0.056 
$3 
2/36 = 5.6% 
$0.168 
$4 
3/36 = 8.3% 
$0.252 
$5 
4/36 = 11.1% 
$0.555 
$6 
5/36 = 13.9% 
$0.833 
$7 
6/36 = 16.7% 
$1.167 
$8 
5/36 = 13.9% 
$1.111 
$9 
4/36 = 11.1% 
$1.000 
$10 
3/36 = 8.3% 
$0.833 
$11 
2/36 = 5.6% 
$0.611 
$12 
1/36 = 2.8% 
$0.333 
Total 
36/36 = 100% 
$7.00 
The expected return? $7.00.

Consider a probability distribution
example with less variability  that is, a distribution
with less risk that your returns deviate from
the average. This time suppose you are throwing
two sixsided dice, but each dice has only three
3's and three 4's  with corresponding returns.
The probability distribution for returns throwing
a pair of these strange dice is
Return 
Probability 
Weighted expected
return 
$6 
9/36 = 25% 
1.500 
$7 
18/36 = 50% 
3.500 
$8 
9/36 = 25% 
2.000 
Total 
36/36 = 100% 
$7.00 
Again, the most likely return is $7 and the expected
return is $7, but now the range is narrower 
from $6 to $8. This smaller variability affects
risk. It is less likely you will have a miserable
return. (Less variability also diminishes the
possibility of a fabulous return.) For a person
who is risk averse, it would be more desirable
to throw the strange 34only dice. There is less
variability, and thus less risk, than with regular
dice.
