A 10 by 10 grid is created using 100 points, as shown. Point $P$ is given. One of the other 99 points is randomly chosen to be $Q$. What is the probability that the line segment $P Q$ is vertical or horizontal?
(A) $\frac{2}{11}$
(B) $\frac{1}{5}$
(C) $\frac{1}{10}$
(D) $\frac{4}{25}$
(E) $\frac{5}{33}$

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-034.jpg?height=366&width=377&top_left_y=701&top_left_x=1340)
Line segment $P Q$ is vertical if $Q$ is chosen from the points in the column in which $P$ lies.

This column contains 9 points other than $P$ which could be chosen to be $Q$ so that $P Q$ is vertical.

Line segment $P Q$ is horizontal if $Q$ is chosen from the points in the row in which $P$ lies. This row contains 9 points other than $P$ which could be chosen to be $Q$ so that $P Q$ is horizontal. Each of these 9 points is different from the 9 points in the column containing $P$.

Thus, there are $9+9=18$ points which may be chosen to be $Q$ so that $P Q$ is vertical or horizontal.

Since there are a total of 99 points to choose $Q$ from, the probability that $Q$ is chosen so that $P Q$ is vertical or horizontal is $\frac{18}{99}$ or $\frac{2}{11}$.

ANSWER: (A)