Let's analyze the problem step-by-step:
1. Consider the starting position: The frog starts at the origin of a coordinate system, which we can denote as [tex]\((0, 0)\)[/tex].
2. Possible directions for the first jump: Cyrus the frog can jump 2 units in either the North, South, East, or West directions.
3. Possible directions for the second jump: After the first jump, the frog can make another 2-unit jump in any direction from its new position.
4. Possible resulting positions:
- From the origin, if the frog jumps 2 units North (to (0, 2)) and then 2 more units East, it lands at (2, 2).
- If it jumps 2 units North and then 2 more units West, it lands at (-2, 2).
- If it jumps 2 units South and then 2 more units East, it lands at (2, -2).
- If it jumps 2 units South and then 2 more units West, it lands at (-2, -2).
- If it jumps 2 units East first (to (2, 0)) and then 2 more units North, it lands at (2, 2).
- If it jumps 2 units East first and then 2 more units South, it lands at (2, -2).
- If it jumps 2 units West first (to (-2, 0)) and then 2 more units North, it lands at (-2, 2).
- If it jumps 2 units West first and then 2 more units South, it lands at (-2, -2).
5. Calculate the distance from the origin: For all the resulting positions, the frog lands at one of the points [tex]\((2, 2)\)[/tex], [tex]\((-2, 2)\)[/tex], [tex]\((2, -2)\)[/tex], or [tex]\((-2, -2)\)[/tex].
- The Euclidean distance from the origin to any of these points can be calculated as follows:
[tex]\[
\sqrt{x^2 + y^2}
\][/tex]
- For [tex]\((2, 2)\)[/tex]:
[tex]\[
\sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.828
\][/tex]
- This calculation shows that the distance to each of these points is approximately 2.828 units.
6. Determine if the frog lands less than 1 unit away from the starting position: Since 2.828 units is the distance from the origin for all possible landing points, and 2.828 units is greater than 1 unit, the frog does not land less than 1 unit away from the starting position in any of the scenarios.
7. Probability calculation: Since the frog has equal probability of jumping in any direction and all possible landing points are definitely more than 1 unit away from the starting position, the probability that the frog lands less than 1 unit away from the origin is 0.
Thus, the probability that Cyrus the frog lands less than 1 unit away from his starting position is:
[tex]\[
\boxed{0}
\][/tex]
