Practical Solutions.pdf

American Roulette has 38 spaces, and various bets can be placed with different payouts and probabilities.
The expected value and standard deviation for each bet type are calculated, with results as follows:
- single number: 35 to 1 payout, 1/38 probability, -€0.53 expected value, 57.62 standard deviation
- single row: 11 to 1 payout, 3/38 probability, -€0.53 expected value, 32.36 standard deviation
- single column: 2 to 1 payout, 12/38 probability, -€0.53 expected value, 13.94 standard deviation
- all red or black numbers: 1 to 1 payout, 18/38 probability, -€0.53 expected value, 9.99 standard deviation
- top line bet: 6 to 1 payout, 5/38 probability, -€0.79 expected value, 23.66 standard deviation

In Dungeons & Dragons, three magical weapons are compared:
- a greatsword with 2d6 + 1 damage: 8 expected value, 2.42 standard deviation
- a flail with 1d10 + 2 damage: 7.5 expected value, 2.87 standard deviation
- a greataxe with 1d12 + 1 damage: 7.5 expected value, 3.45 standard deviation
The greatsword has the highest expected value and lowest standard deviation.

The path diagram encodes structural equations, and the variance of y1 and covariance of y1 and y2 are derived as:
Var(y1) = β1^2 Var(x) + Var(ε1)
Cov(y1, y2) = β1 β2 Var(x)