Probability of hits over long times

Question

A particular event occurs every minute, for instance. When this event starts, it is checked whether a random number (a float between 0.0 to 1.0) is less or equal than 0.1. If so, then an action occurs.

• Every minute: event generate a random number between 0.0 to 1.0;
• If random number is <= to 0.1, do action;
• Else, wait to next minute to try again;

I need know a formula that accept a number of cycles as variable and I receive the possibility to action occur.

• Cycle 1: 10% of chances;
• Cycle n: ...% of chances;
• Cycle 60: ...% of chances;

Answer 1

It solve my problem: Probability that a man will hit the target

Basically, I need reverse my initial possibility (10% will turn 90%) and pow with number of event cycles ("each minute"), so I reverse again to know my possibility:

possibility = 1 - ( 1 - initial possibility ) ^ number of cycles

For instance, for first cycle, considering 10% of initial possibility:

• p = 1 - ( 1 - 0.1 ) ^ 1
• p = 1 - 0.9 ^ 1
• p = 1 - 0.9 = 10% of chances

For 10th cycle, still considering 10% of initial possibility:

• p = 1 - ( 1 - 0.1 ) ^ 10
• p = 1 - 0.9 ^ 10
• p = 1 - 0.349... = 65.1% of chances

Wolfram Alpha reference.