Two coins are tossed. You win \( \$ 6.00 \) if both coins are heads and \( \$ 10.00 \) if both coins are tails. You lose \( \$ 8.00 \) if the coins do not match. What is the expected value of this game? Is this a fair game? The expected value is \( \$ \square \).

To calculate the expected value, we first determine the probabilities of each outcome when two coins are tossed. There are four possible outcomes: HH, HT, TH, and TT. - Probability of HH (both heads) = 1/4, winning $6.00. - Probability of TT (both tails) = 1/4, winning $10.00. - Probability of HT or TH (not matching) = 1/2 (combined), losing $8.00. Now we can calculate the expected value (EV): \[ EV = \left( \frac{1}{4} \times 6 \right) + \left( \frac{1}{4} \times 10 \right) + \left( \frac{1}{2} \times -8 \right) \] \[ EV = 1.5 + 2.5 - 4 = 0 \] So, the expected value is \( \$ 0 \). Since the expected value is zero, this game is fair in terms of potential gain and loss. In pure probability terms, the outcomes are evenly matched against the risks, making you neither gain nor lose money on average in the long run!