1. Determine the total number of unit cubes in the rectangular prism:
$$\text{Total number of unit cubes} = 20 \times 1 \times 7 = 140$$

2. Identify the different types of cubes based on the number of painted faces:
- Corner cubes (3 faces painted):
There are 8 corners in a rectangular prism.
- Edge cubes (2 faces painted):
Each edge has $20 - 2 = 18$ cubes (excluding the corners) for the long edges, and $7 - 2 = 5$ cubes for the short edges. There are 4 long edges and 8 short edges.
$$\text{Total edge cubes} = 4 \times 18 + 8 \times 5 = 72 + 40 = 112$$
- Face cubes (1 face painted):
These are the cubes on the faces but not on the edges or corners. Each face has $20 \times 7 = 140$ cubes, but we need to subtract the edges and corners.
$$\text{Total face cubes} = 140 - 72 - 8 = 60$$
- Interior cubes (0 faces painted):
These are the cubes inside the prism, not on the surface.
$$\text{Total interior cubes} = 140 - (8 + 72 + 60) = 0$$

3. Calculate the probability for each type of cube:
- Corner cubes (3 faces painted):
$$\text{Probability of landing with a painted face up} = \frac{3}{6} = \frac{1}{2}$$
$$\text{Contribution to total probability} = \frac{8}{140} \times \frac{1}{2} = \frac{8}{280} = \frac{1}{35}$$

- Edge cubes (2 faces painted):
$$\text{Probability of landing with a painted face up} = \frac{2}{6} = \frac{1}{3}$$
$$\text{Contribution to total probability} = \frac{72}{140} \times \frac{1}{3} = \frac{72}{420} = \frac{24}{140} = \frac{12}{70} = \frac{6}{35}$$

- Face cubes (1 face painted):
$$\text{Probability of landing with a painted face up} = \frac{1}{6}$$
$$\text{Contribution to total probability} = \frac{60}{140} \times \frac{1}{6} = \frac{60}{840} = \frac{1}{14}$$

4. Sum the contributions to get the total probability:
$$\text{Total probability} = \frac{1}{35} + \frac{6}{35} + \frac{1}{14}$$
Convert $\frac{1}{14}$ to a common denominator:
$$\frac{1}{14} = \frac{2}{28} = \frac{2}{35}$$
$$\text{Total probability} = \frac{1}{35} + \frac{6}{35} + \frac{2}{35} = \frac{9}{35}$$

The final answer is $\boxed{\frac{9}{35}}$.