In the game of​ roulette, a player can place a ​$7 bet on the number 30 and have a StartFraction 1 Over 38 EndFraction
probability of winning. If the metal ball lands on 30​, 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. What is the expected value of the game to the​ player?

Let's break down the problem and calculate the expected value (EV) of the game to the player. The expected value is a measure of what the player can expect to win or lose on average per bet. Here's how we calculate it:

### Step 1: Define the outcomes and their probabilities

1. **Winning outcome**:
   - The player has a $\frac{1}{38}$ probability of winning. 
   - When the player wins, they keep their $7 bet and receive an additional $245, meaning their total gain is $245.

2. **Losing outcome**:
   - The player has a $\frac{37}{38}$ probability of losing.
   - When the player loses, they lose their $7 bet, which means their total loss is $7.

### Step 2: Calculate the expected value for each outcome

The expected value $ EV $ is calculated as:

\[
EV = ( \text{Probability of Winning} \times \text{Winnings} ) + ( \text{Probability of Losing} \times \text{Losses} )
\]

#### Winning outcome:

- Probability of winning: $\frac{1}{38}$
- Total winnings: $245

\[
EV_{\text{win}} = \frac{1}{38} \times 245 = \frac{245}{38} \approx 6.45
\]

#### Losing outcome:

- Probability of losing: $\frac{37}{38}$
- Losses: $7

\[
EV_{\text{loss}} = \frac{37}{38} \times (-7) = \frac{-259}{38} \approx -6.82
\]

### Step 3: Combine the outcomes

Now, combine the two outcomes to find the total expected value:

\[
EV = 6.45 + (-6.82) = -0.37
\]

### Final Expected Value

The expected value of the game for the player is $-0.37$. This means that, on average, the player loses about 37 cents per $7 bet.

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Would you like further clarification on any step or more details? 

Here are some related questions you might explore:
1. How would the expected value change if the payout increased to $300?
2. What is the probability of losing in this game?
3. How much should the payout be to make the game fair for the player?
4. What is the expected loss after playing 100 rounds of the game?
5. What is the probability of losing three times in a row?

**Tip:** The expected value tells you what happens over the long run. A negative expected value suggests a loss is more likely over time, even if a win is possible in the short term.

In the game of​ roulette, a player can place a ​$7 bet on the number 30 and have a 1/38 probability of winning. If the metal ball lands on 30, 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. What is the expected value of the game to the player?

Explore how to calculate the expected value of betting on a single number in roulette with a 1/38 probability of winning and a payout of $245. This step-by-step solution covers probability concepts and expected value calculations, ideal for students in Grades 10-12.

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