The probability of an event happening is $0.55$. What is the probability that the event does not happen?

The sum of the probabilities of an event happening, $P(E)$, and its complement event not happening, $P(E^{\prime })$, is always equal to $1$.
Given:

$$P(E)=0.55$$

Therefore:

$$P(E^{\prime })=1-P(E)$$

$$P(E^{\prime })=1-0.55=0.45$$

The probability that the event does not happen is $0.45$.