A game has an expected value to you of

​$13001300.

It costs

​$13001300

to​ play, but if you​ win, you receive​ $100,000 (including your

​$13001300

​bet) for a net gain of

​$98 comma 70098,700.

What is the probability of​ winning? Would you play this​ game? Discuss the factors that would influence your decision.

Question content area bottom

Part 1

The probability of winning is

enter your response here

​(Type an integer or a​ decimal.)

Part 2

Would you play this​ game? Discuss the factors that would influence your decision.

Since the probability of winning the game is

▼

very low,

very high,

50-50,

it

▼

seems

does not seem

worth it to play the game.

To solve this problem, we need to find the probability of winning based on the given information about expected value and payouts. Here is the detailed breakdown:

### Part 1: Calculating the Probability of Winning

1. **Expected Value Calculation**: The expected value (EV) of the game is given as $1300.
   
2. **Cost to Play**: The cost to play the game is $1300.

3. **Net Gain if Win**: If you win, you receive $100,000 including your $1300 bet, resulting in a net gain of $98,700.

The expected value formula for this game is:

\[
\text{EV} = (P_{\text{win}} \times \text{Net Gain if Win}) + (P_{\text{lose}} \times \text{Net Loss if Lose})
\]

Where:
- $P_{\text{win}}$ is the probability of winning.
- $P_{\text{lose}}$ is the probability of losing, which is $1 - P_{\text{win}}$.
- Net gain if win = $98,700.
- Net loss if lose = -$1300.

Given the expected value (EV) is $1300, we substitute the values into the equation:

\[
1300 = (P_{\text{win}} \times 98700) + [(1 - P_{\text{win}}) \times (-1300)]
\]

Simplifying this:

\[
1300 = 98700P_{\text{win}} - 1300 + 1300P_{\text{win}}
\]

\[
1300 = 100000P_{\text{win}} - 1300
\]

Adding 1300 to both sides:

\[
2600 = 100000P_{\text{win}}
\]

Solving for $P_{\text{win}}$:

\[
P_{\text{win}} = \frac{2600}{100000} = 0.026
\]

### Part 2: Decision to Play the Game

The probability of winning the game is $0.026$ or 2.6%.

#### Would you play this game?

Considering the probability of winning is **very low (2.6%)**, it **does not seem** worth it to play the game.

#### Factors influencing the decision:
1. **Risk Tolerance**: A person with a high tolerance for risk may decide to play despite the low probability of winning.
2. **Financial Situation**: If losing $1300 would significantly impact your finances, it would be prudent not to play.
3. **Alternative Investments**: Consider if there are other ways to invest $1300 that might offer better returns or less risk.
4. **Expected Utility**: Beyond expected value, the utility or satisfaction gained from a potential win versus the loss can influence the decision.
5. **Psychological Factors**: Some people may find excitement or entertainment value in playing the game, which could influence their decision.

Would you like any further details or have any questions?

Here are 5 questions you might ask next:
1. How do you calculate expected value for other types of games?
2. What is the difference between expected value and expected utility?
3. Can you explain how probability theory is applied in real-life scenarios?
4. What are some strategies to manage risk in gambling or investing?
5. How do psychological factors influence decision-making in risky situations?

**Tip**: When considering risky decisions, always weigh the potential losses against the potential gains and consider your personal financial situation and risk tolerance.

Learn how to calculate the probability of winning and expected value in a game problem where the expected value is $1300. Understand the decision-making process based on the calculated probability and net gain. Suitable for advanced high school students and anyone interested in probability and expected value calculations.

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