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Math  /  Data & Statistics

QuestionYou roll a die, winning nothing if the number of spots is odd, $1\$ 1 for a 2 or a 4 , and $10\$ 10 for a 6 . Round your answers to 3 decimal places (a) Find the expected value and standard deviation of your prospective winnings. The expected value is □\square , the standard deviation is □\square (b) You play twice. Find the mean of your total winnings.
The mean is □\square

Studdy Solution

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) :
E(X)=(0×16)+(0×16)+(1×16)+(0×16)+(1×16)+(10×16)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+16+0+16+106=126=2E(X) = 0 + 0 + \frac{1}{6} + 0 + \frac{1}{6} + \frac{10}{6} = \frac{12}{6} = 2

STEP 5

Calculate the variance Var(X) \text{Var}(X) :
Var(X)=E(X2)−(E(X))2\text{Var}(X) = E(X^2) - (E(X))^2
Calculate E(X2) E(X^2) :
E(X2)=(02×16)+(02×16)+(12×16)+(02×16)+(12×16)+(102×16)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(X2)=0+0+16+0+16+1006=1026=17E(X^2) = 0 + 0 + \frac{1}{6} + 0 + \frac{1}{6} + \frac{100}{6} = \frac{102}{6} = 17
Calculate Var(X) \text{Var}(X) :
Var(X)=17−22=17−4=13\text{Var}(X) = 17 - 2^2 = 17 - 4 = 13
Calculate the standard deviation σ(X) \sigma(X) :
σ(X)=Var(X)=13≈3.606\sigma(X) = \sqrt{\text{Var}(X)} = \sqrt{13} \approx 3.606

STEP 6

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

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