To calculate the expected value of the game, we need to determine the probabilities of each outcome and then calculate the expected winnings for each outcome.
Step 1: Determine the total number of marbles.
- Red marbles: 10
- Purple marbles: 6
- Turquoise marbles: 4
Total number of marbles = 10 + 6 + 4 = 20 marbles.
Step 2: Calculate the probability of each type of marble.
- Probability of pulling a red marble: \( P(\text{red}) = \frac{10}{20} = 0.5 \)
- Probability of pulling a purple marble: \( P(\text{purple}) = \frac{6}{20} = 0.3 \)
- Probability of pulling a turquoise marble: \( P(\text{turquoise}) = \frac{4}{20} = 0.2 \)
Step 3: Calculate the expected winnings for each outcome.
- If you pull a red marble, you win $0.
- If you pull a purple marble, you win $2.
- If you pull a turquoise marble, you win $5.
Expected winnings calculation: \[ \text{Expected winnings} = P(\text{red}) \times \text{Winnings if red} + P(\text{purple}) \times \text{Winnings if purple} + P(\text{turquoise}) \times \text{Winnings if turquoise} \] \[ = 0.5 \times 0 + 0.3 \times 2 + 0.2 \times 5 \] \[ = 0 + 0.6 + 1.0 = 1.6 \]
Step 4: Calculate the net expected value. The cost to play the game is $3. Therefore, the expected net winnings (expected value) is given by subtracting the cost to play from the expected winnings: \[ \text{Expected value} = \text{Expected winnings} - \text{Cost to play} \] \[ = 1.6 - 3 = -1.4 \]
The expected value of playing the game is \(\boxed{-1.40}\).