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Let's say I am a slot machine with 100 dollars, if the person next to me has 1000 dollars, would you say this person has a higher chance of winning, lets say, the jackpot? How about any amount of winnings? I was skeptical when I thought of this question because although the person has more money than I do, I always thought each play at a slot machine, or gambling in general, as independent events, but does the fact that he has more money give that person more chances?

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You need to specify the question carefully. If the player plays until either he goes broke or he wins the jackpot, the player with 1000 gets to play on average 10x more, so has (almost) 10x the chance of hitting the jackpot. If he plays until broke, that happens with probability $1$ regardless of the starting capital, even if he wins the jackpot. If he stops for some other reason, you need to specify that.

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