What is the Expected Value of Game 2 (dice) assuming you play all $100 on Game 2?

To calculate the expected value (EV) of a game, we need to know the possible outcomes, their probabilities, and the associated payouts.

Assuming **Game 2** is a dice game, let's clarify a few things. Since I don't have the specific rules for Game 2, I will base the calculation on a typical dice game. Please let me know if the game rules differ.

### Typical Dice Game Example:
Assume:
- You roll a fair 6-sided die, and each face has a different value from 1 to 6.
- The game payout depends on the number rolled (e.g., if you roll 1, the payout is different than rolling 6).
- Let's assume the payouts are linearly related to the dice roll, e.g., rolling a 1 gives a payout of $10, rolling a 2 gives $20, and so on, up to rolling a 6, which gives $60.

### Step 1: Define possible outcomes and payouts
- Roll a 1: Payout = $10
- Roll a 2: Payout = $20
- Roll a 3: Payout = $30
- Roll a 4: Payout = $40
- Roll a 5: Payout = $50
- Roll a 6: Payout = $60

### Step 2: Probability of each outcome
Since it's a fair die, the probability of rolling any number is:
\[
P(\text{roll any number}) = \frac{1}{6}
\]

### Step 3: Calculate the Expected Value (EV)
The expected value is the sum of all possible outcomes multiplied by their respective probabilities. So, the EV of this game would be:

\[
EV = \left( \frac{1}{6} \times 10 \right) + \left( \frac{1}{6} \times 20 \right) + \left( \frac{1}{6} \times 30 \right) + \left( \frac{1}{6} \times 40 \right) + \left( \frac{1}{6} \times 50 \right) + \left( \frac{1}{6} \times 60 \right)
\]

Let's compute this.

\[
EV = \frac{1}{6} \times (10 + 20 + 30 + 40 + 50 + 60)
\]
\[
EV = \frac{1}{6} \times 210 = 35
\]

### Step 4: Apply Your $100 Bet
If you bet $100, the expected payout (based on the EV of 35) would be:

\[
\text{Expected Payout} = \frac{35}{60} \times 100 \approx 58.33
\]

So, if you play all $100 on **Game 2** under these assumptions, the **Expected Value** of your payout is approximately **$58.33**.

---

If the payout structure or rules differ, please clarify, and I can adjust the calculation.

Would you like further details or have any questions?

### Related questions:
1. What is the standard deviation for this game, and how does it affect risk?
2. How does the expected value change if the payouts are different?
3. What is the probability of winning at least $50 in this game?
4. How does the expected value change if you bet half your money?
5. How do multiple rounds affect the expected value over time?

### Tip:
Understanding expected value helps manage risk and maximize long-term gains in probability-based games.

Learn how to calculate the expected value (EV) of Game 2, a dice game, by determining the possible outcomes, their probabilities, and payouts. The calculation involves using basic probability concepts and formulas. Discover how betting $100 can result in an expected payout of approximately $58.33. Suitable for high school students in Grades 10-12.

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