Using \(144\) diamonds, the \(12\) by \(12\) grid of diamonds below is created. One of the diamonds is coloured and labelled \(X.\)
One of the other \(143\) diamonds in the grid is randomly chosen and is coloured in and labelled \(Y\). What is the probability the line segment connecting \(X\) and \(Y\) is vertical or horizontal?
Line segment \(XY\) is vertical if \(Y\) is chosen from the diamonds in the column in which \(X\) lies. In this column there are \(11\) diamonds other than \(X\) which could be chosen to be \(Y\) so that \(XY\) is vertical.
Line segment \(XY\) is horizontal if \(Y\) is chosen from the diamonds in the row in which \(X\) lies. In this row there are \(11\) diamonds other than \(X\) which could be chosen to be \(Y\) so that \(XY\) is horizontal. Each of these \(11\) diamonds is different from the \(11\) diamonds in the column containing \(X.\) Thus, there are \(11+11=22\) diamonds which may be chosen for \(Y\) so that \(XY\) is vertical or horizontal.
Since there are a total of \(143\) diamonds to choose \(Y\) from, the probability that \(Y\) is chosen so that \(XY\) is vertical or horizontal is \(\frac{22}{143}\) or \(\frac{2}{13}.\)