Let's break this down step by step to calculate the expected value of your total winnings after 100 rolls under the optimal strategy.

Step 1: Understand the Rules

1. Initial bet: You pay 2 dollars to play.
2. Dice roll outcomes:
   • Roll 1-3 (Probability = 3/6 = 1/2): You lose all of your money.
   • Roll 4-6 (Probability = 3/6 = 1/2): You get your bet back plus double your bet.

Step 2: Define the Optimal Strategy

• Optimal strategy: You bet the entire amount you have won so far, after each roll, and never end the game early.
• So, after each roll, the amount of money you have is either:
  • Zero (if you roll a 1-3), in which case the game ends, and you lose everything.
  • Twice the current amount (if you roll a 4-6).

Step 3: Model the Problem

Let $X_n$ represent the amount of money you have after the $n$-th roll. Initially, $X_0 = 2$ dollars.

• After the first roll, you either:
  • Lose your entire bet (with probability 1/2), so $X_1 = 0$.
  • Double your bet (with probability 1/2), so $X_1 = 4$ (since you started with 2 dollars).

For each subsequent roll:

• If you have $X_n$ dollars after the $n$-th roll, the amount will change as:
  • $X_{n+1} = 2X_n$ with probability 1/2.
  • $X_{n+1} = 0$ with probability 1/2.

Step 4: Recurrence Relation

The amount of money after $n$ rolls follows a random process where, each time, you either:

• Lose all your money.
• Double your money.

Thus, the value of your total winnings after 100 rolls can be modeled as:

• With probability $(1/2)^{100}$, you lose all your money (because you will have lost in each roll).
• With probability $(1/2)^{99}$, you double your money 100 times, ending with $2^{100}$ dollars.

Step 5: Expected Value of Total Winnings

The expected value $E(X_{100})$ is the average value of the outcome, considering the probabilities of losing all your money or doubling it many times. The expected value is:

$E(X_{100}) = \left(\frac{1}{2}\right)^{100} \times 0 + \left(\frac{1}{2}\right)^{99} \times 2^{100}$

Simplifying this expression:

$E(X_{100}) = \left(\frac{1}{2}\right)^{99} \times 2^{100} = 2$

Final Answer:

The expected value of your total winnings after 100 rolls, following the optimal strategy, is 2 dollars.

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Would you like a more detailed breakdown or further questions on how the probabilities work here? Here are some related questions you might find interesting:

1. How would the expected value change if you were allowed to stop the game at any point?
2. What would happen if the game were structured to pay triple the bet instead of double?
3. How does the number of rolls (100) influence the expected value of the total winnings?
4. What is the variance in your total winnings after 100 rolls, and how would it affect your strategy?
5. How does the probability distribution of outcomes shift if the die has an unfair bias?

Tip: In games like this, a good strategy is often tied to understanding the long-term probabilities rather than short-term outcomes.