What Are the Odds of Winning in Roulette with Different Betting Strategies?

Question

If Alice bets $1 repeatedly on red in roulette and Bob bets $100 once on red, who has a better chance of winning $100?

Answer 1

Bob has a roughly 47.4% chance of winning $100 since he's betting once, while Alice has a significantly lower chance, at 0.00265%, of winning $100 by betting $1 repeatedly due to the need for a higher number of wins over losses.

Comments

Simplest way to think about this, I think, is a smaller bet means you have to win more to reach the total before you walk away.

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I tried an online casino once and chose red 16 times in a row on roulette and it landed on black 16 times. No more.

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I thought of the Alice case via the simulated net profit lines such as at 10:10. It's gonna take her a long time to burn through that $10,000 so probably there will be a streak of  rounds during which she has a gain of $100. Or probably not, when I saw that low probability for Alice's success I realized that getting 100 more wins than losses in some number of rounds might be much more rare than I thought.

For comparison, you can see in the graph at 10:10 a gain of about $25 between rounds 500-600. The trick just is that it should happen pretty much right in the beginning, not when she's already lost some money. The graph I referenced was still losing money after that $25 gain.

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Gut instinct for the problem with Alice and Bob tryna win $100 is that for Alice, you're gonna need to use a hypergeometric distribution, but of course now that I'm writing it out, I realize that I'm probably wrong because for every dollar she wins, that's one more bet that can go wrong without cutting into her progress on trying to win $100, so the total # of trials isn't set

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I feel like this is super intuitive though? Playing multiple games with a less than 50% win rate is obviously gonna be much more unlikely to be less successful than one game with the same rate. How were we supposed to be "tricked" at the beginning?

Like the math is interesting but the premise is obvious, that it will be a very small number for Alice.