Example 3 A uniform cube with the numbers $1,1,2,3,3,5$ marked on its six faces is shown in Figure 3. When this cube is thrown once, the number on the top face is recorded as the x-coordinate of a point in the Cartesian coordinate system, and the number on the bottom face is recorded as the y-coordinate of that point. According to this rule, each throw of the cube yields a point in the plane. Given that the two points Xiao Ming obtained from his first two throws determine a line $l$, and this line $l$ passes through the point $(4,7)$. Then, the probability that the point he gets from his third throw also lies on this line is ( ).
(A) $\frac{2}{3}$
(B) $\frac{1}{2}$
(C) $\frac{1}{3}$
(D) $\frac{1}{6}$
Explanation: Each roll may result in six points with coordinates (two of which are overlapping):
$$
(1,1),(1,1),(2,3),(3,2),(3,5),(5,3) \text {. }
$$

By plotting the points and calculations, it can be found that the line determined by any two of the points $(1,1)$, $(2,3)$, and $(3,5)$ all pass through the point $P(4,7)$. Therefore, the probability that the point obtained by Xiao Ming on his third roll is also on the line $l$ is $\frac{4}{6}=\frac{2}{3}$.

Note: The key to solving this problem lies in listing all the coordinates of the points and determining the equation of the linear function through graphing.