A bag contains 2 gold marbles, 9 silver marbles, and 25 black marbles. Someone offers to play this game: You randomly select one marble from the bag. If it is gold, you win $3. If it is silver, you win $2. If it is black, you lose $1. What is your Expected Value if you play this game? S Round your answer to the nearest cent.
A bag contains 2 gold marbles, 9 silver marbles, and 25 black marbles. Someone offers to play this game: You randomly select one marble from the bag. If it is gold, you win $3. If it is silver, you win $2. If it is black, you lose $1. What is your Expected Value if you play this game? S Round your answer to the nearest cent.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Probability and Expected Value Calculation
A bag contains the following marbles:
- 2 gold marbles
- 9 silver marbles
- 25 black marbles
A game is proposed with the following rules:
- Randomly select one marble from the bag.
- If you select a gold marble, you win $3.
- If you select a silver marble, you win $2.
- If you select a black marble, you lose $1.
**Calculate the Expected Value of playing this game:**
To find the expected value, calculate the probability of each outcome and multiply it by the potential winnings, then sum these values:
1. **Probability of drawing a gold marble:**
\[
P(\text{Gold}) = \frac{2}{2 + 9 + 25} = \frac{2}{36}
\]
2. **Probability of drawing a silver marble:**
\[
P(\text{Silver}) = \frac{9}{36}
\]
3. **Probability of drawing a black marble:**
\[
P(\text{Black}) = \frac{25}{36}
\]
**Calculate Expected Value:**
\[
\text{Expected Value} = (P(\text{Gold}) \times \$3) + (P(\text{Silver}) \times \$2) + (P(\text{Black}) \times -\$1)
\]
Substitute the probabilities:
\[
\text{Expected Value} = \left(\frac{2}{36} \times 3\right) + \left(\frac{9}{36} \times 2\right) + \left(\frac{25}{36} \times -1\right)
\]
**Round your answer to the nearest cent.**
---
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Transcribed Image Text:### Probability and Expected Value Calculation
A bag contains the following marbles:
- 2 gold marbles
- 9 silver marbles
- 25 black marbles
A game is proposed with the following rules:
- Randomly select one marble from the bag.
- If you select a gold marble, you win $3.
- If you select a silver marble, you win $2.
- If you select a black marble, you lose $1.
**Calculate the Expected Value of playing this game:**
To find the expected value, calculate the probability of each outcome and multiply it by the potential winnings, then sum these values:
1. **Probability of drawing a gold marble:**
\[
P(\text{Gold}) = \frac{2}{2 + 9 + 25} = \frac{2}{36}
\]
2. **Probability of drawing a silver marble:**
\[
P(\text{Silver}) = \frac{9}{36}
\]
3. **Probability of drawing a black marble:**
\[
P(\text{Black}) = \frac{25}{36}
\]
**Calculate Expected Value:**
\[
\text{Expected Value} = (P(\text{Gold}) \times \$3) + (P(\text{Silver}) \times \$2) + (P(\text{Black}) \times -\$1)
\]
Substitute the probabilities:
\[
\text{Expected Value} = \left(\frac{2}{36} \times 3\right) + \left(\frac{9}{36} \times 2\right) + \left(\frac{25}{36} \times -1\right)
\]
**Round your answer to the nearest cent.**
---
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