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Imagine that you have a 100-sided die, and you're rolling to determine if event X happens. Event X requires two conditions to be filled; first, you must roll between 1 and 50 on the die (boundaries included). Once you have done that, you must roll a second time, and that result must be between 1 and 60 (boundaries included). So, Roll One must be between 1 and 50, and Roll Two must be between 1 and 60. You have two chances to roll Roll One. That is to say, if your first attempt on Roll One is not 1 through 50, you may take a second attempt. This attempt is taken

*only*if you fail on the first attempt at Roll One. If you succeed at either of these attempts, Roll Two is taken. Roll 2 may only be taken once.

So here's the way I'm looking at this: I'm calling the probability of success on a single roll of Roll One "Y", and the probability of success on Roll Two "Z". I believe that means that the probability of passing Roll One and Roll Two on the first attempt is simply (YZ). From that, I'm calculating the probability of failing Roll One once, but passing Roll One and Roll Two on the reroll to be the failure chance of the first Roll One times the success chance of the reroll: (1-YZ)(YZ). If I'm right so far, then I think the overall chance of success should be the probability of success on the first Roll One times the probability of success on the reroll: (1-YZ)(YZ)^2.

Is this correct? And for bonus points, what would the formula be for calculating the probability of event X being fulfilled N times in a row?

Thanks!