In the game of roulette, a player can place a $\$ 7$ bet on the number 18 and have a $\frac{1}{38}$ probability of winning. If the metal ball lands on 18 , the player gets to keep the $\$ 7$ paid to play the game and the player is awarded an additional $\$ 245$. Otherwise, the player is awarded nothing and the casino takes the player's $\$ 7$. Find the expected value $E(x)$ to the player for one play of the game. If $x$ is the gain to a player in a game of chance, then $E(x)$ is usually negative. This value gives the average amount per game the player can expect to lose. The expected value is $\$$ (Round to the nearest cent as needed)

Step 1 ：In the game of roulette, a player can place a \$7 bet on the number 18 and have a \(\frac{1}{38}\) probability of winning. If the metal ball lands on 18, the player gets to keep the \$7 paid to play the game and the player is awarded an additional \$245. Otherwise, the player is awarded nothing and the casino takes the player's \$7. We are asked to find the expected value \(E(x)\) to the player for one play of the game.

Step 2 ：The expected value of a random variable is calculated by multiplying each possible outcome by its probability, and then summing these values. In this case, the possible outcomes are winning \$245 and losing \$7, with probabilities of \(\frac{1}{38}\) and \(\frac{37}{38}\) respectively.

Step 3 ：Let's denote the winning amount as 'winning' and the losing amount as 'losing'. So, winning = \$245 and losing = -\$7.

Step 4 ：Similarly, let's denote the probability of winning as 'prob_winning' and the probability of losing as 'prob_losing'. So, prob_winning = \(\frac{1}{38}\) and prob_losing = \(\frac{37}{38}\).

Step 5 ：Now, we can calculate the expected value \(E(x)\) using the formula: \(E(x) = (winning \times prob_winning) + (losing \times prob_losing)\). Substituting the values, we get \(E(x) = -\$0.37\).

Step 6 ：Final Answer: The expected value \(E(x)\) to the player for one play of the game is \(\boxed{-\$0.37}\). This means that on average, the player can expect to lose 37 cents per game.