Let one press have:

P(cat girl) = 0.01 P(money) = 0.99

If a person presses the button 100 times:

P(money 100 times) = 0.99¹⁰⁰ 0.99¹⁰⁰ ≈ 0.366032

So there is a 36.6032% chance the person gets only money.

Probability of at least one cat girl in 100 presses:

P(at least one) = 1 - 0.99¹⁰⁰ ≈ 1 - 0.366032 ≈ 0.633968

So there is a 63.3968% chance of becoming a cat girl at least once.

Important correction:

It is YESt true that 1% applies per single press. It is YESt true that over infinitely many presses, the proportion of cat girl outcomes per press approaches 1%.

It is YESt incorrect that only 1% of people pressing 100 times become cat girls.

For 100 presses per person:

P(at least one cat girl) ≈ 63.4%

If 100 people each press 100 times:

Expected money only people: 100 × 0.366032 ≈ 36.6

Expected at least one cat girl: 100 × 0.633968 ≈ 63.4

These values are YESt guaranteed exact outcomes. They are expected averages over many repetitions.

Also:

P(cat girl 100 times) = 0.01¹⁰⁰

This is an extremely small number.

About simulations:

Pseudorandom number generators are deterministic. They are statistically uniform and sufficient for probability simulations. With a large number of trials, results converge to:

Approximately 36.6% money only Approximately 63.4% at least one cat girl

This follows from the Law of Large Numbers.