To calculate the expected winnings from the game, we first need to determine the probabilities for each possible outcome when tossing 2 coins.
When tossing 2 coins, the possible outcomes are:
- Heads & Heads (HH)
- Tails & Tails (TT)
- Heads & Tails (HT)
- Tails & Heads (TH)
Each outcome has a probability of \( \frac{1}{4} \) because there are 4 equally likely outcomes.
Now we can calculate the winnings associated with each outcome:
- For HH (both heads), you win $4.
- For TT (both tails), you win $6.
- For HT (one head, one tail), you lose $8.
- For TH (one tail, one head), you lose $8.
Next, we can calculate the expected value (EV) of the winnings using the formula:
\[ EV = (P(HH) \times W(HH)) + (P(TT) \times W(TT)) + (P(HT) \times W(HT)) + (P(TH) \times W(TH)) \]
Where:
- \( P(HH) = \frac{1}{4}, W(HH) = 4 \)
- \( P(TT) = \frac{1}{4}, W(TT) = 6 \)
- \( P(HT) = \frac{1}{4}, W(HT) = -8 \)
- \( P(TH) = \frac{1}{4}, W(TH) = -8 \)
Now substituting in the values:
\[ EV = \left(\frac{1}{4} \times 4\right) + \left(\frac{1}{4} \times 6\right) + \left(\frac{1}{4} \times -8\right) + \left(\frac{1}{4} \times -8\right) \]
Calculating each term:
\[ EV = 1 + 1.5 - 2 - 2 = 1 + 1.5 - 4 = 2.5 - 4 = -1.5 \]
Thus, the expected winnings are:
\[ \text{Expected Winnings} = -1.5 \]
The answer is \(-1.50\).