In American roulette, the wheel contains the numbers 1 through 36 , alternating between black and red. There are two green spaces numbered 0 and 00 . A player places a bet of $$ \$ 1.75 $$ on red to play the game. If the ball lands on red, the player gets a $$ \$ 1.75 $$ for winning and receives the money back. If the ball does not land on red, then the player simply loses the $$ \$ 1.75 $$ placed on the bet. If the player places the same bet on red 7 times, what is the player's expected winnings? Multiply each pair of values together and add the products to find the player's expected winnings. E(winnings $$ )=\frac{9}{19}(\$ 1.75)+\frac{10}{19}(-\$ 1.75) \approx-\$ 0.09 $$ Identify the possible outcomes of random variable $$ W $$ with respective probabilities of occurring $$ P(W) $$ Write the probability of the ball landing on a red pocket; make sure to

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To solve the problem, we need to identify the possible outcomes of the random variable $$ W $$ (the player's winnings) when betting on red in American roulette, and calculate the expected winnings based on the probabilities of these outcomes.

### Step 1: Identify the possible outcomes

In American roulette, there are 38 total pockets: 18 red, 18 black, and 2 green (0 and 00). When a player bets on red, there are two possible outcomes for each bet:

1. **Winning Outcome**: The ball lands on red.
   - Winnings: $$ \$ 1.75 $$ (the player wins back their bet plus the winnings).
   - Probability of this outcome: $$ P(W = 1.75) = \frac{18}{38} = \frac{9}{19} $$.

2. **Losing Outcome**: The ball does not land on red (it lands on black or green).
   - Winnings: $$ -\$ 1.75 $$ (the player loses their bet).
   - Probability of this outcome: $$ P(W = -1.75) = \frac{20}{38} = \frac{10}{19} $$.

### Step 2: Calculate the expected winnings for one bet

The expected winnings $$ E(W) $$ for one bet can be calculated using the formula:

$$
E(W) = P(W = 1.75) \cdot 1.75 + P(W = -1.75) \cdot (-1.75)
$$

Substituting the probabilities:

$$
E(W) = \frac{9}{19} \cdot 1.75 + \frac{10}{19} \cdot (-1.75)
$$

### Step 3: Calculate the expected winnings

Now, let's perform the calculations:

1. Calculate $$ \frac{9}{19} \cdot 1.75 $$:
   $$
   \frac{9}{19} \cdot 1.75 = \frac{9 \cdot 1.75}{19} = \frac{15.75}{19}
   $$

2. Calculate $$ \frac{10}{19} \cdot (-1.75) $$:
   $$
   \frac{10}{19} \cdot (-1.75) = \frac{-10 \cdot 1.75}{19} = \frac{-17.5}{19}
   $$

3. Combine the results:
   $$
   E(W) = \frac{15.75}{19} + \frac{-17.5}{19} = \frac{15.75 - 17.5}{19} = \frac{-1.75}{19}
   $$

4. Calculate the final expected winnings:
   $$
   E(W) \approx -0.0921
   $$

### Step 4: Expected winnings for 7 bets

Since the player places the same bet on red 7 times, the expected winnings for 7 bets would be:

$$
E(W_{7}) = 7 \cdot E(W) = 7 \cdot \frac{-1.75}{19}
$$

Calculating this:

$$
E(W_{7}) = \frac{7 \cdot -1.75}{19} = \frac{-12.25}{19} \approx -0.6447
$$

### Conclusion

The player's expected winnings after placing the same bet on red 7 times is approximately $$ -\$ 0.64 $$. This means that, on average, the player can expect to lose about 64 cents after 7 bets on red in American roulette.
After placing the same bet on red 7 times, the player's expected winnings are approximately a loss of \$0.64.

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