select the correct answer.\na game involves rolling a fair six - sided die. if the number facing upward on…

select the correct answer.\na game involves rolling a fair six - sided die. if the number facing upward on the die is a whole number multiple of three, the player wins an amount equal to the number on the die times $20. if the number is not a multiple of three, the player gets nothing. what is the expected value of a players winnings on each roll?\n\na. $3.33\nb. $6.66\nc. $8.50\nd. $30.00\ne. $40.00

select the correct answer.\na game involves rolling a fair six - sided die. if the number facing upward on the die is a whole number multiple of three, the player wins an amount equal to the number on the die times $20. if the number is not a multiple of three, the player gets nothing. what is the expected value of a players winnings on each roll?\n\na. $3.33\nb. $6.66\nc. $8.50\nd. $30.00\ne. $40.00

Answer

Explanation:

Step1: Identify winning numbers and probabilities

The numbers on a six - sided die are 1, 2, 3, 4, 5, 6. The multiples of three are 3 and 6. The probability of rolling a 3 or 6 on a fair six - sided die is $P(3)=\frac{1}{6}$ and $P(6)=\frac{1}{6}$, and the probability of not getting a multiple of three $P(\text{not 3 or 6})=\frac{4}{6}$. When the number is not a multiple of three, the winning amount $X_1 = 0$. When the number is 3, the winning amount $X_2=3\times20 = 60$ and when the number is 6, the winning amount $X_3 = 6\times20=120$.

Step2: Use the expected - value formula

The expected - value formula is $E(X)=\sum_{i}x_ip_i$. Here, $E(X)=0\times\frac{4}{6}+60\times\frac{1}{6}+120\times\frac{1}{6}$. First, calculate $0\times\frac{4}{6}=0$. Second, calculate $60\times\frac{1}{6} = 10$. Third, calculate $120\times\frac{1}{6}=20$. Then $E(X)=0 + 10+20=30$.

Answer:

D. $30.00