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I have a Big Dreams lottery and a Gold Rush lottery in my Festival Plaza. The Big Dreams is a higher rank, and yet to date I have won 5 big nuggets, and have yet to win a Master Ball. I assume the odds of winning the grand prize is different for each kind of lottery based on my experience, is this correct?

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Well, you’re sort of correct.

In Festival Plaza, there are 3 different lottery shops: Big Dreams, Gold Rush, and Treasure Hunt. Each of the 3 lotteries have 10 different items to win. The 3rd-10th item to win is the same for all three booths, but the 1st and 2nd prizes vary based on what shop you have. Also, the rank of the booth does play an effect - where in a 1-star rank, you may find yourself getting a lot of Berry Juice and Full Heal, but in a 5-star rank, it's much easier to win the top prizes.

So, to answer your question, the different lottery booths don't effect your outcome, but instead the types of prizes you win. The rank of the booth is what adjusts the odds, where 1-star and 5-star being the lowest and highest rank respectively. You may refer to the this or this link if you need more clarification, as well as seeing the prizes they have to offer.

Hope I helped! :)

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Already knew all that, good sir. What I cannot explain is how I have had better fortune from a 3-star Gold Rush than a 5-star Big Dreams. Higher rank would indicate greater chance of winning better prizes if all three different lotteries weighted the odds the same way.
Well, the top prizes in a 5-star rank does not mean 10% chance each, it could simply just change from maybe 1 to 5%. Maybe you were just luckier in Gold Rush, because all should be weighted the same.
I've won 5 big nuggets and no Master Balls, I suspect that it's not just luck.
When you install a lottery for the first time, its random seed is always set to the same value so you will always get the same prize on the first play. Notably, any Treasure Hunt other than level 1 will always give you prize #2 (a bottle cap) on the first play, and any Gold Rush other than level 1 will always give you prize **#1** (a big nugget) on the first play. However, there is no level of Big Dreams for which the first-play prize is one of the top two tiers, so you have to wait for at least your second drawing to have any chance whatsoever of winning a Rare Candy or Master Ball from that.