Expected Value Example



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