Bold Betting Strategy

Suppose you have 3 tokens for a betting game, and your goal is to increase your fortune to 5 tokens before running out. You devise a bold strategy where, on every turn, you bet as many tokens as possible against the house, but not more than necessary to reach 5 tokens. You always have a probability of 1/3 to win the bet.

• If you bet X tokens and lose, you lose the X tokens.
• If you win, the house adds X tokens to your balance.
• Example: if you have 2 tokens and you bet both of them, and you win, then your balance increased to 2 + 2 = 4 tokens. Compute the probability that you reach 5 tokens before running out.

Answer format: round to 3 decimal places.