Solution:

Adding chords one at a time, each new chord increases the region count by $1$ plus the number of existing chords it crosses inside the disk. Starting from one region, the expected total is $1 + 27 + E,$ where $E$ is the expected number of interior crossing pairs. The two diameters cross once. A random chord's endpoints land in one of the $6$ quadrant pairs, each with probability $\frac{1}{6}.$ The chord crosses the vertical diameter exactly when its endpoints have opposite $x$-signs, which happens for $4$ of the $6$ pairs, so it meets each diameter with probability $\frac{2}{3}$ and both diameters together $\frac{4}{3}$ times on average: the $25$ chords contribute $\frac{100}{3}$ expected crossings with the diameters.

For two random chords, condition on their quadrant pairs ($36$ equally likely ordered combinations). If one uses quadrants $1, 3$ and the other $2, 4,$ the endpoints always alternate, so they always cross: $2$ combinations. If the two pairs are adjacent and disjoint, such as $\{1, 2\}$ and $\{3, 4\},$ the chords never cross: $4$ combinations. In each of the other $30$ combinations, whether the endpoints alternate around the circle reduces to comparing independent uniform points inside shared quadrants — for example, a $\{1,3\}$ chord and a $\{1,2\}$ chord cross exactly when the two quadrant-$1$ points come in one specific order — and the probability is $\frac{1}{2}$ by symmetry. So two random chords cross with probability $$\frac{2 \cdot 1 + 4 \cdot 0 + 30 \cdot \frac{1}{2}}{36} = \frac{17}{36}.$$

The $\binom{25}{2} = 300$ chord pairs contribute $300 \cdot \frac{17}{36} = \frac{425}{3}$ expected crossings, so $E = 1 + \frac{100}{3} + \frac{425}{3} = 176$ and the expected number of regions is $1 + 27 + 176 = 204.$