What is the Expected Value of Game 1 (blackjack), assuming you play all $100 on Game 1?

To compute the **expected value** (EV) of a game, we need to know the possible outcomes and their associated probabilities. For **Game 1 (blackjack)**, we assume you're referring to a standard setup where you're playing blackjack at a casino with $100.

In blackjack, there are several possible outcomes that affect the EV:

1. **Winning the hand**: If you win, you typically get paid **1:1** on your bet (i.e., you win the amount of your bet, $100).
2. **Losing the hand**: If you lose, you lose your $100 bet.
3. **Tying the hand (push)**: If there's a tie, you don't lose or win anything; your $100 is returned to you.
4. **Blackjack (21 on the first two cards)**: If you hit a blackjack, the typical payout is **3:2** (i.e., you win 1.5 times your bet, which is $150).

To calculate the EV, we must factor in the **probabilities** of each outcome.

Let's break it down:

1. **Win Probability**: Typically, the player's win probability in blackjack is around **42%**.
2. **Lose Probability**: The player's losing probability is around **49%**.
3. **Push Probability**: The probability of a tie (push) is around **9%**.
4. **Blackjack Probability**: The probability of getting a blackjack is typically around **4.75%** (this is included in the win probability).

### Expected Value Formula:

\[
EV = (P_{\text{win}} \times \text{payout}_{\text{win}}) + (P_{\text{blackjack}} \times \text{payout}_{\text{blackjack}}) + (P_{\text{push}} \times 0) + (P_{\text{lose}} \times -\text{bet})
\]

Where:

- $P_{\text{win}} = 42\%$
- $P_{\text{blackjack}} = 4.75\%$
- $P_{\text{push}} = 9\%$
- $P_{\text{lose}} = 49\%$
- Payout for a win is $100.
- Payout for a blackjack is $150.
- The bet is $100.

Let's calculate it:

\[
EV = (0.42 \times 100) + (0.0475 \times 150) + (0.49 \times -100)
\]

I'll compute this value for you.The **expected value** (EV) of playing $100 on Game 1 (blackjack) is **$0.125**. This means that, on average, you can expect to gain 12.5 cents per game played over the long run, assuming these probabilities hold.

Would you like more details, or do you have any other questions?

Here are 5 related questions you could explore:
1. How do card counting strategies affect the expected value in blackjack?
2. What is the house edge in blackjack, and how does it compare to other casino games?
3. How does changing the bet size affect the expected value in blackjack?
4. What is the impact of different rule variations (e.g., dealer stands on soft 17) on the EV?
5. How does the number of decks used in blackjack affect the expected value?

**Tip:** Always understand the house edge before playing any casino game—it helps you make informed decisions.

Explore the expected value of playing blackjack with a $100 bet. This detailed solution covers the probabilities of winning, losing, pushing, and hitting blackjack, along with the relevant mathematical concepts, such as expected value and payout probabilities. Suitable for students in Grades 10-12.

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