Backing Out the Late-Delivery Rate

A logistics firm records, for every parcel it ships, whether the parcel arrives late. Let

$X = \begin{cases} 1 & \text{the parcel is late} \\ 0 & \text{the parcel is on time} \end{cases}$

so that $X \sim \text{Bernoulli}(p)$, where $p$ is the probability that a randomly chosen parcel is late.

An internal audit reports that

$\text{Var}(X) = 0.24$

and it is also known that fewer than half of all parcels arrive late.

What is $p$?

Round to two decimal places.