To determine the probability that Alice is exactly $2\sqrt{2}$ units away from the origin after 4 seconds, we need to consider the geometry and combinatorics of her random walk.

Step 1: Geometric interpretation

The distance $2\sqrt{2}$ from the origin corresponds to points that satisfy: $x^2 + y^2 = (2\sqrt{2})^2 = 8.$ This equation describes all points on a circle with radius $2\sqrt{2}$ centered at the origin. However, Alice moves in steps of 1 unit, so her reachable points lie on a lattice (integer coordinates). For $x^2 + y^2 = 8$, the lattice points are: $(x, y) = (\pm 2, \pm 2).$ Thus, there are 4 lattice points: $(2, 2), (-2, 2), (-2, -2), (2, -2)$.

Step 2: Valid paths

Alice takes 4 steps, each either up, down, left, or right. To reach $(2, 2)$, for example, she must:

1. Move 2 steps right (R) and 2 steps up (U), in any order.

The number of such arrangements is: $\binom{4}{2, 2} = \frac{4!}{2! \cdot 2!} = 6.$ The same logic applies to the other 3 lattice points $(-2, 2), (-2, -2), (2, -2)$. Each point can be reached in 6 ways.

Step 3: Total possible paths

Alice can take $4$ steps, and for each step, there are 4 choices (up, down, left, right). Thus, the total number of possible paths is: $4^4 = 256.$

Step 4: Probability calculation

There are $6 \times 4 = 24$ favorable paths that result in Alice being $2\sqrt{2}$ units away from the origin. Hence, the probability is: $\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{24}{256} = \frac{3}{32}.$

Final Answer:

The probability that Alice is exactly $2\sqrt{2}$ units away from the origin after 4 seconds is: $\boxed{\frac{3}{32}}$

---

Do you want more details on any step or need further clarification?
Here are 5 related questions for further exploration:

1. How would the problem change if Alice could only move diagonally?
2. What if Alice is restricted to 6 steps instead of 4?
3. How can you compute probabilities for other distances, such as $\sqrt{2}$ or $3\sqrt{2}$?
4. How does the probability change if Alice chooses directions with unequal probabilities?
5. How can this problem be modeled using generating functions or recurrence relations?

Tip: When solving random walk problems, visualize the reachable lattice points to simplify calculations.