Introduction: In this problem, we consider a random walker on a square lattice where the walker can take four steps in total. The probability that the walker is at a site which is only 2 steps away is to be determined.

Approach:

To solve this problem, we can consider the possible paths that the walker can take to reach a site which is only 2 steps away.

There are two possible ways to reach a site which is only 2 steps away:
1. The walker takes two steps in one direction, and then two steps in a perpendicular direction.
2. The walker takes one step in one direction, then turns 90 degrees and takes another step, then turns 180 degrees and takes a third step, and finally turns 90 degrees again and takes a fourth step.

Let us consider each of these cases separately:

Case 1: The walker takes two steps in one direction, and then two steps in a perpendicular direction.

In this case, there are two possible directions in which the walker can take the first two steps, and two possible directions in which the walker can take the second two steps. Therefore, there are a total of 4 possible paths that the walker can take to reach a site which is only 2 steps away.

The probability of taking any one of these paths is $(1/4)^4$, since each step has a probability of 1/4 of being taken. Therefore, the total probability of reaching a site which is only 2 steps away via Case 1 is:

$$4 \times (1/4)^4 = 1/64$$

Case 2: The walker takes one step in one direction, then turns 90 degrees and takes another step, then turns 180 degrees and takes a third step, and finally turns 90 degrees again and takes a fourth step.

In this case, there are four possible directions in which the walker can take the first step, and two possible directions in which the walker can take the second step (since it must be perpendicular to the first step). Once the first two steps have been taken, the direction of the next two steps is uniquely determined.

Therefore, there are a total of 4 x 2 = 8 possible paths that the walker can take to reach a site which is only 2 steps away via Case 2.

The probability of taking any one of these paths is $(1/4)^4$, since each step has a probability of 1/4 of being taken. Therefore, the total probability of reaching a site which is only 2 steps away via Case 2 is:

$$8 \times (1/4)^4 = 1/32$$

Final Probability: The total probability of reaching a site which is only 2 steps away is the sum of the probabilities obtained from Case 1 and Case 2:

$$1/64 + 1/32 = 3/128$$

Therefore, rounding off to two decimal places, the probability is between 0.44 and 0.48.