Math Snap

STEP 1

1. A fair six-sided die is rolled.
2. The winnings are defined as: $0 for rolling 1, 3, or 5; $1 for rolling 2 or 4; and $10 for rolling 6.
3. The probabilities for each outcome are equal, $\frac{1}{6}$.

STEP 2

1. Calculate the expected value of a single roll.
2. Calculate the variance and standard deviation of a single roll.
3. Calculate the mean of total winnings when playing twice.

STEP 3

List the possible outcomes and their probabilities:
- Roll 1, 3, or 5: Winnings = $0, Probability = $\frac{1}{6}$ each
- Roll 2 or 4: Winnings = $1, Probability = $\frac{1}{6}$ each
- Roll 6: Winnings = $10, Probability = $\frac{1}{6}$

STEP 4

Calculate the expected value $E(X)$:
$$E(X) = (0 \times \frac{1}{6}) + (0 \times \frac{1}{6}) + (1 \times \frac{1}{6}) + (0 \times \frac{1}{6}) + (1 \times \frac{1}{6}) + (10 \times \frac{1}{6})$$ $$E(X) = 0 + 0 + \frac{1}{6} + 0 + \frac{1}{6} + \frac{10}{6} = \frac{12}{6} = 2$$

STEP 5

Calculate the variance $\text{Var}(X)$:
$$\text{Var}(X) = E(X^2) - (E(X))^2$$ Calculate $E(X^2)$:
$$E(X^2) = (0^2 \times \frac{1}{6}) + (0^2 \times \frac{1}{6}) + (1^2 \times \frac{1}{6}) + (0^2 \times \frac{1}{6}) + (1^2 \times \frac{1}{6}) + (10^2 \times \frac{1}{6})$$ $$E(X^2) = 0 + 0 + \frac{1}{6} + 0 + \frac{1}{6} + \frac{100}{6} = \frac{102}{6} = 17$$ Calculate $\text{Var}(X)$:
$$\text{Var}(X) = 17 - 2^2 = 17 - 4 = 13$$ Calculate the standard deviation $\sigma(X)$:
$$\sigma(X) = \sqrt{\text{Var}(X)} = \sqrt{13} \approx 3.606$$

Calculate the mean of total winnings for two plays:
The expected value for one roll is 2, so for two rolls:
$$\text{Mean of total winnings} = 2 + 2 = 4$$ The expected value is:
$$\boxed{2}$$ The standard deviation is:
$$\boxed{3.606}$$ The mean of total winnings for two plays is:
$$\boxed{4}$$