Bold Or Timid At Bad Odds
Kenny has 3 dollars and wants to leave with 6. He wins each round with probability and loses with probability 3/4, independently, and a win pays even money on whatever he staked.
Two candidate strategies:
- Bold: stake all 3 dollars on a single round. He doubles up with probability 1/4.
- Timid: stake 1 dollar per round until he is broke or holds 6.
What is the probability that the TIMID strategy gets him to 6 dollars?
Answer format: round to 3 decimal places.
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