+64 The grid below is made up of line segments, like the line segment in red.
[asy]
unitsize(1 cm);
int i;
for (i = 0; i <= 3; ++i) {
draw((i,0)--(i,2));
}
for (i = 0; i <= 2; ++i) {
draw((0,i)--(3,i));
}
draw((3,1)--(3,2), linewidth(2*bp) + red);
label("$A$", (0,0), SW);
label("$B$", (3,2), NE);
[/asy]
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 paths that pass through it?
+415 Certainly, let's find the expected number of paths that pass through a randomly chosen line segment in the 3x2 grid.
1. Count the Total Number of Paths
To go from 'a' to 'b', we need to make 3 rightward moves and 2 upward moves.
The total number of paths is given by the combination formula: C(5, 3) = 5! / (3! * 2!) = 10
2. Count Paths Through a Horizontal Line Segment
Let's consider a horizontal line segment within the grid.
To reach this segment, we need to make a specific number of rightward moves.
Let's say the segment is reached after 'r' rightward moves.
Then, to reach 'b' from this segment, we need to make (3 - r) rightward moves and 2 upward moves.
The number of paths passing through this segment is: C(r + 2, r) * C(5 - (r + 2), 3 - r) = C(r + 2, r) * C(3 - r, 3 - r) = C(r + 2, r)
Summing this over all possible positions of the horizontal segment (r = 0, 1, 2): Total paths through horizontal segments = C(2, 0) + C(3, 1) + C(4, 2) = 1 + 3 + 6 = 10
3. Count Paths Through a Vertical Line Segment
Similarly, for a vertical line segment reached after 'u' upward moves: Number of paths = C(u + 3, u)
Summing over all possible positions of the vertical segment (u = 0, 1): Total paths through vertical segments = C(3, 0) + C(4, 1) = 1 + 4 = 5
4. Total Paths Through All Line Segments
Total paths through all line segments = 10 (horizontal) + 5 (vertical) = 15
5. Calculate Expected Number of Paths
There are a total of 11 line segments in the grid (6 horizontal and 5 vertical).
Expected number of paths through a randomly chosen line segment = Total paths through all line segments / Total number of line segments
Expected number of paths = 15 / 11
Therefore, the expected number of paths that pass through a randomly chosen line segment in the grid is 15/11.
