The problem gives us a circle with the equation $x^2 + y^2 = 16$, meaning it's centered at the origin $(0, 0)$ with a radius of 4. The task is to find the probability that a randomly chosen integer point $(x, y)$ within or on the circle (but no further than 4 units from the center) has a distance between 3 and 4 from the center.

Steps to solve:

1. Circle equation: The equation $x^2 + y^2 = 16$ represents a circle with radius 4. All points with integer coordinates that satisfy $x^2 + y^2 \leq 16$ are within this circle.

2. Region of interest: We are asked to find the probability that the distance of a point is between 3 and 4. This means we are looking for points that satisfy $9 < x^2 + y^2 \leq 16$, since $\text{distance} = \sqrt{x^2 + y^2}$, and the distance should be more than 3 but less than or equal to 4.

3. Finding integer solutions:

   • We first find the integer points satisfying $x^2 + y^2 \leq 16$ (all integer points within the full circle).
   • Then, we find the integer points that satisfy $9 < x^2 + y^2 \leq 16$ (points between the two circles with radii 3 and 4).

4. Counting the points:

   • For $x^2 + y^2 = 16$ (points exactly on the circle with radius 4): The solutions include points like $(4, 0), (0, 4), (-4, 0), (0, -4), (3, 1), (1, 3), (-1, 3), etc.$.
   • For $9 < x^2 + y^2 \leq 16$, we similarly calculate which points lie between these boundaries.

5. Calculating probability:

   • Let $N_{\text{total}}$ be the total number of integer points within or on the circle with radius 4.
   • Let $N_{\text{region}}$ be the number of integer points within the region $9 < x^2 + y^2 \leq 16$.
   • The probability is then $\frac{N_{\text{region}}}{N_{\text{total}}}$.

I will now calculate the exact integer points and probability.The total number of integer points within or on the circle with radius 4 is 49. Out of these, 20 points have a distance between 3 and 4 from the center. Therefore, the probability that a randomly chosen point has a distance more than 3 but less than or equal to 4 is approximately $0.408$ or 40.82%.

Would you like further details or clarification? Here are 5 related questions you might find useful:

1. How do you determine the integer points within any circle?
2. What is the general formula for the equation of a circle in Cartesian coordinates?
3. How can we apply probability concepts to geometric shapes?
4. What are other applications of this type of probability problem in real life?
5. How would the probability change if we used non-integer points?

Tip: When dealing with geometric probabilities, symmetry can often simplify the process of counting or integrating over specific regions.