Tony is playing a game in which he spins a spinner with 6 equal-sized slices numbered 1 through 6. The spinner stops on a numbered slice at random. This game is this: Tony spins the spinner once. He wins $1 if the spinner stops on the number 1, $3 if the spinner stops on the number 2, $5 if the spinner stops on the number 3, and $7 if the spinner stops on the number 4. He loses $8 if the spinner stops on 5 or 6. (a) Find the expected value of playing the game. dollars (b) What can Tony expect in the long run, after playing the game many times? O Tony can expect to gain money. He can expect to win dollars per spin. O Tony can expect to lose money. He can expect to lose dollars per spin. O Tony can expect to break even (neither gain nor lose money). 5 ?
Tony is playing a game in which he spins a spinner with 6 equal-sized slices numbered 1 through 6. The spinner stops on a numbered slice at random. This game is this: Tony spins the spinner once. He wins $1 if the spinner stops on the number 1, $3 if the spinner stops on the number 2, $5 if the spinner stops on the number 3, and $7 if the spinner stops on the number 4. He loses $8 if the spinner stops on 5 or 6. (a) Find the expected value of playing the game. dollars (b) What can Tony expect in the long run, after playing the game many times? O Tony can expect to gain money. He can expect to win dollars per spin. O Tony can expect to lose money. He can expect to lose dollars per spin. O Tony can expect to break even (neither gain nor lose money). 5 ?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![**Title: Expected Value in Probability Games**
**Tony's Spinner Game: Understanding Expected Value**
Tony is playing a game in which he spins a spinner with 6 equal-sized slices numbered 1 through 6. The spinner stops on a numbered slice at random.
**The Game Rules:**
- Tony spins the spinner once.
- He wins $1 if the spinner stops on the number 1.
- He wins $3 if the spinner stops on the number 2.
- He wins $5 if the spinner stops on the number 3.
- He wins $7 if the spinner stops on the number 4.
- He loses $8 if the spinner stops on 5 or 6.
**Problem Analysis:**
(a) **Find the expected value of playing the game.**
\[ \text{Expected value} = \_ \_ \_ \_ \_ \text{ dollars} \]
(b) **What can Tony expect in the long run, after playing the game many times?**
- Tony can expect to gain money. He can expect to win \_ \_ \_ \_ \_ dollars per spin.
- Tony can expect to lose money. He can expect to lose \_ \_ \_ \_ \_ dollars per spin.
- Tony can expect to break even (neither gain nor lose money).
**Calculation:**
To determine the expected value, we multiply each outcome by its probability and sum the results.
Here are the possible winnings and their corresponding probabilities:
1. Win $1: \(\frac{1}{6}\)
2. Win $3: \(\frac{1}{6}\)
3. Win $5: \(\frac{1}{6}\)
4. Win $7: \(\frac{1}{6}\)
5. Lose $8: \(\frac{2}{6}\)
The expected value (E) can be calculated using the formula:
\[ E = \left(1 \times \frac{1}{6}\right) + \left(3 \times \frac{1}{6}\right) + \left(5 \times \frac{1}{6}\right) + \left(7 \times \frac{1}{6}\right) + \left(-8 \times \frac{2}{6}\right) \]
This educational module explores how understanding expected value in probability can illuminate the long-term outcomes of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F424746dd-29d3-476a-868c-5a06ada19580%2Fb69726de-f131-4421-a55c-6e20801220b5%2Fgp77uqv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Expected Value in Probability Games**
**Tony's Spinner Game: Understanding Expected Value**
Tony is playing a game in which he spins a spinner with 6 equal-sized slices numbered 1 through 6. The spinner stops on a numbered slice at random.
**The Game Rules:**
- Tony spins the spinner once.
- He wins $1 if the spinner stops on the number 1.
- He wins $3 if the spinner stops on the number 2.
- He wins $5 if the spinner stops on the number 3.
- He wins $7 if the spinner stops on the number 4.
- He loses $8 if the spinner stops on 5 or 6.
**Problem Analysis:**
(a) **Find the expected value of playing the game.**
\[ \text{Expected value} = \_ \_ \_ \_ \_ \text{ dollars} \]
(b) **What can Tony expect in the long run, after playing the game many times?**
- Tony can expect to gain money. He can expect to win \_ \_ \_ \_ \_ dollars per spin.
- Tony can expect to lose money. He can expect to lose \_ \_ \_ \_ \_ dollars per spin.
- Tony can expect to break even (neither gain nor lose money).
**Calculation:**
To determine the expected value, we multiply each outcome by its probability and sum the results.
Here are the possible winnings and their corresponding probabilities:
1. Win $1: \(\frac{1}{6}\)
2. Win $3: \(\frac{1}{6}\)
3. Win $5: \(\frac{1}{6}\)
4. Win $7: \(\frac{1}{6}\)
5. Lose $8: \(\frac{2}{6}\)
The expected value (E) can be calculated using the formula:
\[ E = \left(1 \times \frac{1}{6}\right) + \left(3 \times \frac{1}{6}\right) + \left(5 \times \frac{1}{6}\right) + \left(7 \times \frac{1}{6}\right) + \left(-8 \times \frac{2}{6}\right) \]
This educational module explores how understanding expected value in probability can illuminate the long-term outcomes of
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