A game of chance is played on a wheel that contains 38 numbers (18 are red, 18 are black, and 2 are green). When the wheel is spun, the ball is equally likely to land on any of the 38 numbers. Suppose that you bet $9 on black. If the ball lands on a black number, you win $9; otherwise you lose your $9. Let The expected value is. X be the amount you win on your $9 bet. Then X is the random variable whose probability distribution is as follows. a. Find the expected value of the random variable X. (Round to three decimal places as needed.) b. On average, how much will you lose per play? 9 -9 P(X = x) 0.474 0.526 You will lose an average of $ per play. (Round to three decimal places as needed.) Use this to complete parts (a) through (d) to the right. c. Approximately how much would you expect to lose if you bet $9 on black 100 times? 1000 times If you bet $9 on black 100 times you would expect to lose $ (Round to the nearest ten cents as needed.) If you bet $9 on black 1000 times you would expect to lose $ (Round to the nearest dollar as needed.) d. Is the game profitable to play? Explain. Choose the correct answer below. O A. No, because the expected value of X represents the expected profit for this game. A negative number means the game is not profitable. O B. Yes, because the average profit per game is positive. O C. No, the game is not profitable if you can lose money by playing it. Each time you play, you can lose $9. O D. Yes, the game is profitable because you can make a profit playing the game. Each time you
A game of chance is played on a wheel that contains 38 numbers (18 are red, 18 are black, and 2 are green). When the wheel is spun, the ball is equally likely to land on any of the 38 numbers. Suppose that you bet $9 on black. If the ball lands on a black number, you win $9; otherwise you lose your $9. Let The expected value is. X be the amount you win on your $9 bet. Then X is the random variable whose probability distribution is as follows. a. Find the expected value of the random variable X. (Round to three decimal places as needed.) b. On average, how much will you lose per play? 9 -9 P(X = x) 0.474 0.526 You will lose an average of $ per play. (Round to three decimal places as needed.) Use this to complete parts (a) through (d) to the right. c. Approximately how much would you expect to lose if you bet $9 on black 100 times? 1000 times If you bet $9 on black 100 times you would expect to lose $ (Round to the nearest ten cents as needed.) If you bet $9 on black 1000 times you would expect to lose $ (Round to the nearest dollar as needed.) d. Is the game profitable to play? Explain. Choose the correct answer below. O A. No, because the expected value of X represents the expected profit for this game. A negative number means the game is not profitable. O B. Yes, because the average profit per game is positive. O C. No, the game is not profitable if you can lose money by playing it. Each time you play, you can lose $9. O D. Yes, the game is profitable because you can make a profit playing the game. Each time you
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Understanding the Probability Game and Expected Values
A game of chance is played on a wheel with 38 numbers: 18 red, 18 black, and 2 green. When spun, the ball has an equal chance of landing on any number. In this game, a $9 bet on black wins $9 if successful, and loses $9 otherwise.
#### Probability Distribution Table
| x | 9 | -9 |
|------|-------|-------|
| P(X=x) | 0.474 | 0.526 |
- **x:** Outcome based on your $9 bet.
- **P(X=x):** Probability of each outcome.
Use this information to analyze the following questions:
#### Part (a): Expected Value of X
- **Objective:** Calculate the expected value of the random variable X.
- **Formula:** \(E(X) = \sum [x \cdot P(X=x)]\)
#### Part (b): Average Loss Per Play
- **Objective:** Determine the average loss for each play of the game.
- **Interpretation:** This value represents your potential profit or loss over time.
#### Part (c): Cumulative Loss Over Multiple Plays
- **Objective:** Estimate your total loss if you play 100 or 1000 times.
- **Practical Use:** Helps understand long-term outcomes of repeated plays.
#### Part (d): Game Profitability
- **Objective:** Decide if the game is worth playing based on expected outcomes.
- **Options:**
- A: No, negative expected value indicates unprofitability.
- B: Yes, a positive average profit is present.
- C: No, potential for $9 loss each play.
- D: Yes, possibility of profit in every game.
This analysis is crucial for understanding risk and reward in probability-based games.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff534fa3a-7fc6-4517-a79f-1e1d7853b104%2F3312bbb2-b87f-4386-8aec-42c6a2abf0fe%2F1hibgh_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding the Probability Game and Expected Values
A game of chance is played on a wheel with 38 numbers: 18 red, 18 black, and 2 green. When spun, the ball has an equal chance of landing on any number. In this game, a $9 bet on black wins $9 if successful, and loses $9 otherwise.
#### Probability Distribution Table
| x | 9 | -9 |
|------|-------|-------|
| P(X=x) | 0.474 | 0.526 |
- **x:** Outcome based on your $9 bet.
- **P(X=x):** Probability of each outcome.
Use this information to analyze the following questions:
#### Part (a): Expected Value of X
- **Objective:** Calculate the expected value of the random variable X.
- **Formula:** \(E(X) = \sum [x \cdot P(X=x)]\)
#### Part (b): Average Loss Per Play
- **Objective:** Determine the average loss for each play of the game.
- **Interpretation:** This value represents your potential profit or loss over time.
#### Part (c): Cumulative Loss Over Multiple Plays
- **Objective:** Estimate your total loss if you play 100 or 1000 times.
- **Practical Use:** Helps understand long-term outcomes of repeated plays.
#### Part (d): Game Profitability
- **Objective:** Decide if the game is worth playing based on expected outcomes.
- **Options:**
- A: No, negative expected value indicates unprofitability.
- B: Yes, a positive average profit is present.
- C: No, potential for $9 loss each play.
- D: Yes, possibility of profit in every game.
This analysis is crucial for understanding risk and reward in probability-based games.
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