Sometimes a poor gambler could hit a 1000x bet with a $1 bet and win $1000 which he could keep and stop gambling for the week because it is a lot for him. Meanwhile, a rich person could win the same bet and continue gambling because a thousand dollars is nothing for him. He might continue betting for more multipliers and they could lose all the winnings. So, it doesn't matter! It depends on gamblers behaviour.
No, the example you give is incorrect because a rich man does not make $1 bets.
In general the rich tend to bet less than the poor, and that is a fact that can be seen in the fact that where there are more bookies is in the poorer neighbourhoods of the cities and where more lotteries are sold as well. But when the rich bet, they bet in relation to their wealth, and they will not bet $1. They will bet $100 or $1,000 or whatever depending on their wealth.