Example
During one holiday season, the Texas lottery played a game called the Stocking Stuffer. Shown here are the various prizes and the probability of winning each prize. Compute the expected value of the game.
|
Prize (x) |
Probability P(x) |
|
$1,000 |
.00002 |
|
100 |
.00063 |
|
20 |
.00400 |
|
10 |
.00601 |
|
4 |
.02403 |
|
2 |
.08877 |
|
1 |
.10479 |
|
0 |
.77175 |
|
Prize (x) |
Probability P(x) |
x . P(x) |
|
$1,000 |
0.00002 |
0.02000 |
|
100 |
0.00063 |
0.06300 |
|
20 |
0.00400 |
0.08000 |
|
10 |
0.00601 |
0.06010 |
|
4 |
0.02403 |
0.09612 |
|
2 |
0.08877 |
0.17754 |
|
1 |
0.10479 |
0.10479 |
|
0 |
0.77175 |
0.00000 |
|
|
∑[x . P(x)] = |
0.60155 |
|
µ = E(x) = ∑[x . P(x)] = |
0.60155 |
|
The mean, µ = 0.60155.
The expected payoff for a $1 ticket in this game is 60.2 cents. In the long run, the participant will lose about $1.00 – 0.602 = 0.398 or about 40 cents.
Using the mean, the variance:
2
= ∑[(x - µ)2 
P(x)]
and standard deviation:
= 
can be computed.
|
Prize (x) |
Probability P(x) |
(x - µ)2 |
(x - µ)2 . P(x) |
|
$1,000 |
0.00002 |
998797.26186 |
19.97595 |
|
100 |
0.00063 |
9880.05186 |
6.22443 |
|
20 |
0.00400 |
376.29986 |
1.50520 |
|
10 |
0.00601 |
88.33086 |
0.53087 |
|
4 |
0.02403 |
11.54946 |
0.27753 |
|
2 |
0.08877 |
1.95566 |
0.17360 |
|
1 |
0.10479 |
0.15876 |
0.01664 |
|
0 |
0.77175 |
0.36186 |
0.27927 |
|
µ = |
0.60155 |
|
|
2
= ∑[(x - µ)2 P(x)]
= 28.98349 (dollar)2
= = $ 5.38363
