Probability of using the same key

Question

A cipher has 365 different keys. Each day, a key is chosen (uniformly) at random from the 365 keys. After how many days is the chance of any key being reused greater than 50%?

Answer 1

Unless I am reading incorrectly or missing something, the chance of any key is at 0.2% at the beginning. Getting a repeat would only decrease the probability.