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

The **expected value** (EV) is a concept from probability that helps determine the average outcome of a game or event over a long period. For a game like blackjack, the expected value is based on the player's strategy, the rules of the game, and the probability distribution of outcomes (winning, losing, or tying). 

Let’s break down the calculation in terms of a simplified version of Game 2 (blackjack), assuming the following:

- You **bet $100** on each game.
- You either **win**, **lose**, or **push** (tie).

For the purpose of calculating the **expected value** (EV) of Game 2, we need the following probabilities:

1. **P(win)**: The probability of winning the game.
2. **P(lose)**: The probability of losing the game.
3. **P(push)**: The probability of tying the game.

Now, the outcomes are typically:

- **If you win**: You win your $100 bet plus an additional amount, depending on the payout. In standard blackjack, a winning hand pays out at 1:1, and a blackjack hand (Ace + 10-point card) often pays out at 3:2. For simplicity, we assume a 1:1 payout here.
- **If you lose**: You lose your $100.
- **If you push**: You neither win nor lose, resulting in a net gain of $0.

The **Expected Value (EV)** formula is:

\[
EV = (P(\text{win}) \times \text{win payout}) + (P(\text{push}) \times 0) + (P(\text{lose}) \times -100)
\]

To proceed, I would need the **probabilities of win, push, and loss** for this game, as they depend on the specific blackjack rules and strategies used.

### General Calculation (Example Probabilities)
Let's assume the following probabilities based on common blackjack odds:

- **P(win) = 42% (0.42)**
- **P(push) = 8% (0.08)**
- **P(lose) = 50% (0.50)**

Using the formula:

\[
EV = (0.42 \times 100) + (0.08 \times 0) + (0.50 \times -100)
\]

\[
EV = 42 + 0 - 50 = -8
\]

So, the **expected value is -$8**. This means that, on average, for every $100 bet on Game 2 (blackjack), the player would lose $8 over time, assuming these probabilities and the rules of the game.

Would you like more details on any of the steps or assumptions made in this calculation? 

### Related Questions:
1. How do different blackjack rule variations affect the expected value?
2. What is the effect of different payout ratios on the expected value?
3. How does the use of card counting strategies impact the expected value?
4. What is the impact of house rules on the player's winning probability?
5. How does the expected value change if you play multiple rounds instead of one?

**Tip**: Managing your bankroll is essential in games like blackjack. Always bet amounts you can afford to lose since the game has a negative expected value.

Understand how to calculate the expected value of a $100 bet in blackjack using probabilities of winning, losing, and pushing. Learn how the expected value formula is applied in this real-life probability problem. Suitable for Grades 10-12.

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