Monica tosses a fair 6-sided die.  If the roll is a prime number, then she wins that amount of dollars (so that, for example, if she rolls 3, then she wins 3 dollars).  If the roll is composite, she wins nothing. Otherwise, she loses 3 dollars. What is the expected value of her winnings on one die toss? Express your answer as a dollar value to the nearest cent.
The only primes on a die are 2, 3, and 5. The only composites are 4 and 6. The only other option is rolling a 1. There is a $\dfrac{1}{6}$ probability that she will roll a 1 and lose $\$3$, a $\dfrac{1}{3}$ probability of rolling a composite and winning $\$0$, and a $\dfrac{1}{6}$ probability of winning each of $\$2$, $\$3$, or $\$5$.  So $$E = \dfrac{1}{3}\times \$0 + \dfrac{1}{6}\times(\$2+\$3+\$5) + \dfrac{1}{6} \times -\$3 \approx \boxed{\$1.17}.$$
1.17
Level 4
Counting & Probability
Problem: Monica tosses a fair 6-sided die.  If the roll is a prime number, then she wins that amount of dollars (so that, for example, if she rolls 3, then she wins 3 dollars).  If the roll is composite, she wins nothing. Otherwise, she loses 3 dollars. What is the expected value of her winnings on one die toss? Express your answer as a dollar value to the nearest cent.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the expected value of Monica's winnings, we first identify the possible outcomes when rolling a fair 6-sided die. The die can land on the numbers 1, 2, 3, 4, 5, or 6. 

Next, we categorize these numbers based on their mathematical properties:
- Prime numbers: 2, 3, 5
- Composite numbers: 4, 6
- Neither prime nor composite: 1

Monica's winnings are determined as follows:
- If she rolls a prime number (2, 3, or 5), she wins that amount in dollars.
- If she rolls a composite number (4 or 6), she wins nothing.
- If she rolls a 1, she loses 3 dollars.

Now, we calculate the winnings for each outcome:
- Rolling a 1: Winnings = -3 dollars
- Rolling a 2: Winnings = 2 dollars
- Rolling a 3: Winnings = 3 dollars
- Rolling a 4: Winnings = 0 dollars
- Rolling a 5: Winnings = 5 dollars
- Rolling a 6: Winnings = 0 dollars

Next, we determine the probability of each outcome. Since the die is fair, each number has a probability of 1/6.

Now we compute the expected value (EV) using the formula:
EV = (Probability of outcome) x (Winnings for that outcome)

Calculating the expected value for each outcome:
- For rolling a 1: EV = (1/6) * (-3) = -0.5
- For rolling a 2: EV = (1/6) * 2 = 0.3333
- For rolling a 3: EV = (1/6) * 3 = 0.5
- For rolling a 4: EV = (1/6) * 0 = 0
- For rolling a 5: EV = (1/6) * 5 = 0.8333
- For rolling a 6: EV = (1/6) * 0 = 0

Now, we sum these expected values:
Total EV = -0.5 + 0.3333 + 0.5 + 0 + 0.8333 + 0
Total EV = 1.1666...

Rounding this to the nearest cent gives us $1.17.<|END_OF_LATENT|> The final answer is $1.17$.