An exhibition was held in a hall on $15$ August $2022$ between $\text{3 PM}$ and $\text{4 PM}$ during which any person was allowed to enter only once. Visitors who entered before $\text{3:40 PM}$ exited the hall exactly after $20$ minutes from their time of entry. Visitors who entered at or after $\text{3:40 PM}$, exited exactly at $4 \mathrm{~PM}$. The probability distribution of the arrival time of any visitor is uniform between $3 \mathrm{~PM}$ and $4 \mathrm{~PM}$. Two persons $X$ and $Y$ entered the exhibition hall independent of each other. Which one of the following values is the probability that their visits to the exhibition overlapped with each other?
- $\frac{5}{9}$
- $\frac{4}{9}$
- $\frac{2}{9}$
- $\frac{7}{9}$