In the diagram, it is only possible to travel along an edge in the direction indicated by the arrow. Hazel studied the figure, and determined all the possible routes from $A$ to $B$. She selected one of these routes at random. What is the probability that she selected a route which passes through $X$ ?
(A) $\frac{8}{11}$
(B) $\frac{3}{11}$
(C) 1
(D) $\frac{9}{11}$
(E) $\frac{6}{11}$

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-123.jpg?height=263&width=396&top_left_y=1256&top_left_x=1317)
We determine all of the possible routes:

Travelling $A$ to $X$ to $B$, there are 2 routes, since there are 2 paths $A$ to $X$.

Travelling $A$ to $X$ to $Y$ to $B$, there are 6 routes since there are two paths from $A$ to $X$ and 3 paths from $Y$ to $B$.

Travelling $A$ to $Y$ to $B$, there are 3 routes, since there are 3 paths $Y$ to $B$.

So there are 11 routes from $A$ to $B$, of which 8 pass through $X$.

Therefore, the probability that Hazel chooses a route that passes through $X$ is $\frac{8}{11}$.

ANSWER: (A)