Many casinos have a game called the Big Six Money Wheel, which has 54 slots in which are displayed a Joker, the casino logo, and various dollar amounts, as shown in the table at the top of the next column. Players may bet on the Joker, the casino logo, or one or more dollar denominations. The wheel is spun and if the wheel stops on the same place as the player's bet, the player wins that amount for each dollar bet. Denomination	Number of slots
$40 (Joker)	1
$40 (Casino logo)	1
$20	2
$10	4
$5	7
$2	15
$1	24
If a player bets $1 on the Casino logo denomination, find the player's expectation. (Round your answer to two decimal places.)

In order to calculate the player's expectation when betting $1 on the Casino logo denomination, we can use the formula for expected value (EV) in probability:

\[
\text{EV} = \sum (\text{Payout} \times \text{Probability of the outcome})
\]

### Step-by-Step Breakdown:

1. **Determine the Payout**:
   - The payout for landing on the Casino logo is $40 (the amount won for a $1 bet).

2. **Determine the Probability**:
   - The total number of slots on the wheel is 54.
   - The Casino logo occupies only 1 of these slots.
   - Therefore, the probability of landing on the Casino logo is:

   \[
   \text{Probability of Casino logo} = \frac{1}{54}
   \]

3. **Determine the Loss Amount**:
   - If the wheel does not land on the Casino logo, the player loses their $1 bet.
   - The probability of losing (i.e., landing on any other slot) is:

   \[
   \text{Probability of losing} = \frac{53}{54}
   \]

4. **Calculate the Expected Value**:
   - The player wins $40 with a probability of $\frac{1}{54}$, and loses $1 with a probability of $\frac{53}{54}$.
   - The expected value (EV) is:

   \[
   \text{EV} = (40 \times \frac{1}{54}) + (-1 \times \frac{53}{54})
   \]

   Simplifying this:

   \[
   \text{EV} = \frac{40}{54} - \frac{53}{54}
   \]

   \[
   \text{EV} = \frac{40 - 53}{54} = \frac{-13}{54}
   \]

   \[
   \text{EV} \approx -0.24
   \]

### Conclusion:

The player's expected value (EV) for betting $1 on the Casino logo denomination is approximately **-0.24**. This means that, on average, the player loses 24 cents for every $1 bet.

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Would you like further details, or do you have any questions?

Here are 5 related questions to explore:
1. What is the expected value of betting on the Joker?
2. How does the expected value change when betting on the $1 denomination?
3. How can the payout odds impact the player's strategy in this game?
4. What is the probability of landing on any denomination with a payout of $20 or more?
5. How would the expected value change if the number of slots on the wheel were different?

**Tip**: In probability games, understanding the expected value can help you gauge how favorable the game is for the player versus the house.

If a player bets $1 on the Casino logo denomination, find the player's expectation. (Round your answer to two decimal places.)

Calculate the expected value of betting $1 on the Casino logo in the Big Six Money Wheel game. Understand the probabilities and payouts involved, and learn how to compute the expected value using the formula for probability. This example is ideal for students in Grades 10-12.

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