Question

we will answer this problem in parts. you play the following game with your sister: you roll a 6 - sided die, and if the result is greater than 4, she gives you $5. if the result is 4 or less, you give her $1. we want to find the expected value of this game. first, for the two types of outcomes to this experiment, enter the probability of each as a fraction: - rolling a 5 or 6: type your answer. - rolling a 1 through 4: type your answer. next, find the expected value of each outcome by multiplying each probability by the value for that outcome. (that is, multiply its probability by how much money you would make if that outcome occurs. round to nearest cent, and do not write the dollar sign.) - rolling a 5 or 6: choose your answer. - rolling a 1 through 4: choose your answer. find the expected value of this game by adding your two previous answers. choose your answer.

Explanation:

Step1: Calculate probabilities

The probability of rolling a 5 or 6 on a 6 - sided die: There are 2 favorable outcomes (5 and 6) out of 6 total outcomes. So the probability $P(5\text{ or }6)=\frac{2}{6}=\frac{1}{3}$. The probability of rolling a 1 through 4: There are 4 favorable outcomes (1, 2, 3, 4) out of 6 total outcomes. So the probability $P(1 - 4)=\frac{4}{6}=\frac{2}{3}$.

Step2: Calculate expected value of each outcome

If you roll a 5 or 6, you get $5. So the expected - value of this outcome is $E_1 = P(5\text{ or }6)\times5=\frac{1}{3}\times5=\frac{5}{3}\approx1.67$. If you roll a 1 through 4, you give $1. So the expected - value of this outcome is $E_2=P(1 - 4)\times(- 1)=\frac{2}{3}\times(-1)=-\frac{2}{3}\approx - 0.67$.

Step3: Calculate the expected value of the game

The expected value of the game $E = E_1+E_2=\frac{5}{3}-\frac{2}{3}=1$.

Answer:

• rolling a 5 or 6: $\frac{1}{3}$
• rolling a 1 through 4: $\frac{2}{3}$
• rolling a 5 or 6 (expected value): $1.67$
• rolling a 1 through 4 (expected value): $-0.67$
• expected value of the game: $1$