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I have this problem that I am trying to solve:

There is an event with a 1.83% chance or ~0.0183 probability of occurring. There is also a score counter that starts with a score of 100. Samples (or cycles) are run with a 100-1.83=98.17% chance of the event NOT occurring. Every sample where the event doesn't occur scores you +1. When the event occurs, it scores you -31 from the total score.

My question is, considering that these events can take place multiple times, how would one estimate the chance or probability of the score being >= 0 after 100, 200, 500 or N number of samples? A step by step explanation would be very appreciated if possible!

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    Here is a start to how you could figure it out for 100 samples. Note that the special event would need to happen at least $7$ times in order to give you a score less than $0$ in the end. So you can find $P(X \lt 6)$ for the binomial distribution with $n=100, p=0.0183$ where X represents the number of special events occurring. – 2017-02-09
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    Thank you for the comment, I have simulated binomial distribution graph in Matlab and approximated the probability for that! – 2017-02-09

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