You are at a Casino and your grandfather gave you $100 to play at the casino and you can’t use the money for anything else.

There are two games you can play.

Game one is blackjack.  Each hand costs $1 to play.  For each hand, you have a 49% chance of winning $2 and a 51% chance of winning nothing.

Game two is dice.  Each play costs $1.  The dice has the numbers 1-6 and you are equally likely to land on each number.  If you roll a 1, you win $0.3.  If you roll a 2, you win $0.5. If you roll a 3, you win $0.7. If you roll a 4, you win $0.9. If you roll a 5, you win $1.10. If you roll a 6, you win $1.3.

What is your expected value for each game, assuming you would play all $100 on the game.

Which game would you choose to play, and why?

To determine which game is more favorable to play, we need to calculate the expected value (EV) for each game.

### Game 1: Blackjack

In blackjack:
- Each hand costs $1 to play.
- You have a 49% chance of winning $2.
- You have a 51% chance of winning nothing.

The expected value for each hand of blackjack can be calculated as follows:

\[
EV_{\text{blackjack}} = (0.49 \times 2) + (0.51 \times 0) = 0.49 \times 2 = 0.98
\]

Thus, the expected value per hand of blackjack is $0.98.

If you play 100 hands, the total expected value is:

\[
EV_{\text{total\ blackjack}} = 100 \times 0.98 = 98
\]

### Game 2: Dice

In the dice game:
- Each roll costs $1 to play.
- The outcomes and winnings are as follows:
  - Rolling a 1: win $0.3
  - Rolling a 2: win $0.5
  - Rolling a 3: win $0.7
  - Rolling a 4: win $0.9
  - Rolling a 5: win $1.10
  - Rolling a 6: win $1.3

The expected value for each roll of the dice can be calculated by considering the probabilities and outcomes:

\[
EV_{\text{dice}} = \left(\frac{1}{6} \times 0.3\right) + \left(\frac{1}{6} \times 0.5\right) + \left(\frac{1}{6} \times 0.7\right) + \left(\frac{1}{6} \times 0.9\right) + \left(\frac{1}{6} \times 1.10\right) + \left(\frac{1}{6} \times 1.3\right)
\]

\[
EV_{\text{dice}} = \frac{1}{6} \left(0.3 + 0.5 + 0.7 + 0.9 + 1.10 + 1.3\right)
\]

\[
EV_{\text{dice}} = \frac{1}{6} \times 4.8 = 0.8
\]

Thus, the expected value per roll of the dice is $0.80.

If you play 100 rolls, the total expected value is:

\[
EV_{\text{total\ dice}} = 100 \times 0.80 = 80
\]

### Comparison and Decision

- **Blackjack Expected Total Value**: $98
- **Dice Expected Total Value**: $80

Based on the expected values, **blackjack** has a higher expected value ($98) compared to the dice game ($80). Therefore, you should choose to play blackjack, as it offers a better return on the $100 you have to play with.

### Summary

- Expected value per hand of blackjack: $0.98
- Expected value per roll of dice: $0.80
- Total expected value for 100 hands of blackjack: $98
- Total expected value for 100 rolls of dice: $80
- **Recommended game to play: Blackjack**

Would you like more details or have any questions?

Here are some related questions:

1. How do probabilities affect the expected value in gambling games?
2. What is variance, and how does it impact gambling outcomes?
3. How would playing fewer hands or rolls change the expected values?
4. Can you explain the concept of risk versus reward in the context of these games?
5. What strategies can be employed to maximize winnings in blackjack?
6. How does the house edge influence long-term outcomes in casino games?
7. What are some common misconceptions about probability in gambling?
8. How would changing the payout structure of the dice game affect its expected value?

**Tip:** Always understand the rules and probabilities of any gambling game you play to make informed decisions about where to place your bets.

Explore the expected values of two casino games, blackjack and dice, to determine which game offers a better return on investment. Understand the probabilities and calculations involved in each game. Recommended for adults interested in probability and gambling strategies.

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