"1 success in 50 trials" is no good. There can be 50 failures in 50 trials and the first still will be int eh first 50.

Your second try is also no good, not because you don't have it right, but because you don't understand it.

Who cares what the distribution is. Think it through.

Pr(1st failure = 1) = 0.01

Pr(1st failure = 2) = 0.01 * 0.99

Pr(1st failure = 3) = 0.01 * 0.99^2

Pr(1st failure = 4) = 0.01 * 0.99^3

Pr(1st failure = 5) = 0.01 * 0.99^4

Pr(1st failure = 6) = 0.01 * 0.99^5

...

Pr(1st failure = 50) = 0.01 * 0.99^49

Can you add those up?

0.01 * (1 + 0.99 + 0.99^2 + ... + 0.99^49)

I hope so.

If, after solving the problem, you recognize the distribution - more power to you. If you really want to play with properties, find the distribution and play with them. If you really want to solve the problem, think it through - don't go shopping for distribution names.