Question

Determine whether the following statement makes sense or does not make sense, and explain your reasoning.

Here's my dilemma, I can accept a $1400 bill or play a game ten times. For each roll of the single die, I win $500 for rolling 1 or 2 ; I win $300 for rolling 3 ; and I lose $100 for rolling 4 , 5, or 6. Based on the expected value, I should accept the $1400 bill.
Choose the correct answer below, and fill in the answer box to complete your choice.
(Round to the nearest cent as needed. Do not include the \$ symbol in your answer.)
A. The statement makes sense because the expected value after ten rolls is dollars, which is less than the value of the current bill.
B. The statement does not make sense because the expected value after ten rolls is dollars, which is greater than the value of the current bill.

Answer 1

The expected value after ten rolls of the die in the described game is $1666.70, which is higher than $1400. Therefore, the statement does not make sense because it is financially better to play the game based on its expected value.

Step-by-step explanation:

To determine whether the statement makes sense, we need to calculate the expected value of playing the game ten times and compare it to the guaranteed $1400 bill. The expected value is calculated by multiplying each outcome by its probability and summing up these amounts.

The expected value for a single roll of the die is calculated as follows:

1. Winning $500 with a probability of 1/3 (rolling a 1 or 2)
2. Winning $300 with a probability of 1/6 (rolling a 3)
3. Losing $100 with a probability of 1/2 (rolling a 4, 5, or 6)

The expected value for a single roll (EV) is:
EV = (1/3 * $500) + (1/6 * $300) + (1/2 * -$100) = ($166.67) + ($50) + (-$50) = $166.67

Therefore, the expected value after ten rolls would be ten times the expected value of a single roll, which is $166.67 * 10 = $1666.70. Since this amount is greater than $1400, the statement does not make sense.

The correct choice is:

B. The statement does not make sense because the expected value after ten rolls is 1666.70 dollars, which is greater than the value of the current bill.

Answer 2

The expected value of playing the game ten times is $1666.70, which is greater than the $1400 bill. Hence, the statement does not make sense since the expected outcome of the game is more financially favorable.

Step-by-step explanation:

To find out whether the statement makes sense, we need to calculate the expected value (EV) of playing the game ten times. The EV for a single die roll in the game is calculated as follows:

• P(win $500) = Probability of rolling a 1 or 2 = 2/6
• P(win $300) = Probability of rolling a 3 = 1/6
• P(lose $100) = Probability of rolling a 4, 5 or 6 = 3/6

The EV for one roll is:

(2/6) × $500 + (1/6) × $300 + (3/6) × (-$100) = ($166.67 + $50 - $50) = $166.67

Thus, the EV for ten rolls is 10 × $166.67 = $1666.70.

Comparing the EV after ten rolls to the value of the $1400 bill shows that the EV is more than $1400. Therefore, the student should choose to play the game instead of accepting the $1400 bill.

The correct answer is:
B. The statement does not make sense because the expected value after ten rolls is 1666.70 dollars, which is greater than the value of the current bill.