The grid below is made up of line segments, like the line segment in red. There are a number of paths that go from A to B in the grid, where every step goes to the right or up. If we choose a line segment at random, then what is the expected number of squares that it is next to?
To determine the expected number of squares that a line segment is next to, we need to calculate the probability of each possible neighboring square and multiply it by the number of squares that have that probability.
Let's analyze the grid starting from an arbitrary position and calculate the number of neighboring squares for each segment in the grid.
In the figure below, we represent the number of neighboring squares for each segment using different colors:
1 (red)
2 (green)
3 (blue)
2 (green)
1 (red)
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| | | |
| R | G | G |
| | | |
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| | | |
| R | B | B |
| | | |
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| | | |
| R | G | G |
| | | |
---------------------
| | | |
| R | G | G |
| | | |
---------------------
| | | |
| R | G | G |
| | | |
---------------------
Based on the grid structure, we can deduce the following:
- Segments on the corners have 2 neighboring squares.
- Segments on the edges (excluding corners) have 3 neighboring squares.
- Segments in the interior have 4 neighboring squares.
Counting the number of each type of segment, we find:
- There are 4 corner segments.
- There are 12 edge segments.
- There are 9 interior segments.
Calculating the expected number of neighboring squares:
- Corner segments contribute 2 * 4 = 8 neighboring squares.
- Edge segments contribute 3 * 12 = 36 neighboring squares.
- Interior segments contribute 4 * 9 = 36 neighboring squares.
Adding up the contributions, we get a total of 8 + 36 + 36 = 80 neighboring squares.
Since there are a total of 30 segments in the grid (4 corners + 12 edges + 9 interior segments), the expected number of squares that a line segment is next to is 80 / 30 ≈ 2.67.
Therefore, the expected number of squares that a line segment is next to is approximately 2.67.
To find the expected number of squares that a randomly chosen line segment is next to, we need to calculate the probability of the line segment being adjacent to each square and then multiply it by the number of adjacent squares.
Let's analyze the grid:
```
A - ─ │ ─ │ ─ │ ─ │ ─ │ ─
│ . │ · │ · │ · │ · │ ·
└ ─ │ ─ │ ─ │ ─ │ ─ │
│ . │ · │ · │ · │
└ ─ │ ─ │ ─ │ ─ │ ─ │
│ . │ · │ · │
└ ─ │ ─ │ ─ │ ─ │
│ . │ · │
└ ─ │ ─ │ ─ │
│ . │
└ ─ │ ─ │
B
```
Consider the square adjacent to the chosen line segment (marked as "·" in the grid). The line segment can be adjacent to at most two squares (up to four for the corners). So, there are eight segments that touch a given square.
Now, let's calculate the probability of a line segment being adjacent to a square in the grid.
We start from point A and want to reach point B through an uninterrupted path of line segments. The number of squares that the path touches is equal to the Manhattan distance between A and B (number of horizontal steps + number of vertical steps).
In this case, the Manhattan distance between A and B is 4 (2 horizontal steps + 2 vertical steps), so the path touches 4 squares.
Since there are 8 line segments touching each square, the probability of a randomly chosen line segment being adjacent to a square is 4/8 = 1/2.
Finally, multiply the probability (1/2) by the number of adjacent squares (4) to find the expected number of squares that a randomly chosen line segment is next to:
Expected number = Probability * Number of Adjacent Squares
= (1/2) * 4
= 2
Therefore, the expected number of squares that a randomly chosen line segment is next to is 2.
To find the expected number of squares that a randomly chosen line segment is next to, we need to consider the probability of each possible outcome.
Let's analyze the grid and its line segments. Suppose the grid has M rows and N columns.
For a line segment that is not on the boundary of the grid, it will be next to 4 squares.
For a line segment on the boundary of the grid, there are three cases to consider:
1. Corner segments:
- For the four corner segments, they are next to only 2 squares.
2. Edge segments (non-corner):
- For the segments on the top and bottom rows (excluding corners), they are next to 3 squares.
- For the segments on the left and right columns (excluding corners), they are also next to 3 squares.
- Therefore, the total number of edge segments is 2(M-2) + 2(N-2).
Now, let's calculate the expected number of squares that a randomly chosen line segment is next to.
The probability of selecting a line segment not on the boundary is:
P(not on boundary) = (M-2)(N-2) / (M*N)
The expected number of squares for this case is:
E(not on boundary) = P(not on boundary) * 4
The probability of selecting a corner segment is:
P(corner) = 4 / (M*N)
The expected number of squares for corner segments is:
E(corner) = P(corner) * 2
The probability of selecting an edge segment is:
P(edge) = (2(M-2) + 2(N-2)) / (M*N)
The expected number of squares for edge segments is:
E(edge) = P(edge) * 3
Now, we can calculate the overall expected number of squares:
Expected number of squares = E(not on boundary) + E(corner) + E(edge)
Expected number of squares = [((M-2)(N-2) / (M*N)) * 4] + [(4 / (M*N)) * 2] + [((2(M-2) + 2(N-2)) / (M*N)) * 3]
Simplifying this expression will give you the final answer for the expected number of squares that a randomly chosen line segment is next to.