L-Shaped Room

Answer: $\frac{8}{9}$98​

We split the room into 3 unit squares: A, B, C, where B is the centre square connecting the other two.

Case 1: Both people in the same square

If both people are in the same square, they can always see each other. The combinations are AA, BB, CC.

The probability of both being in the same square is $\frac{1}{3}$31​.

Case 2: At least one person in square B

If at least one person is in square B (the centre), they can always see each other—square B has direct line of sight to both A and C.

The combinations are AB, BA, BC, CB (excluding BB, which we already counted).

The probability is $\frac{4}{9}$94​.

Case 3: People in opposite squares (A and C)

The remaining probability is $\frac{2}{9}$92​ for the combinations AC and CA.

In this case, whether they can see each other depends on the line of sight between them. We consider adding conceptual walls around square B. If the line of sight passes through square B’s internal walls, they can see each other. If it passes through the L-shape’s outer walls, they cannot.

By symmetry, there’s an equal chance of the line of sight passing through square B’s walls or the outer walls.

This means there’s a $\frac{1}{2}$21​ chance within the $\frac{2}{9}$92​ case that they can see each other—contributing $\frac{1}{9}$91​ to the total probability.

Total probability