Part of a City
Source:
January 19, 2006
geometryrectangleprobabilitysymmetry
Problem Statement
The adjacent map is part of a city: the small rectangles are rocks, and the paths in between are streets. Each morning, a student walks from intersection A to intersection B, always walking along streets shown, and always going east or south. For variety, at each intersection where he has a choice, he chooses with probability whether to go east or south. Find the probability that through any given morning, he goes through . [asy]
defaultpen(linewidth(0.7)+fontsize(8));
size(250);
path p=origin--(5,0)--(5,3)--(0,3)--cycle;
path q=(5,19)--(6,19)--(6,20)--(5,20)--cycle;
int i,j;
for(i=0; i<5; i=i+1) {
for(j=0; j<6; j=j+1) {
draw(shift(6*i, 4*j)*p);
}}
clip((4,2)--(25,2)--(25,21)--(4,21)--cycle);
fill(q^^shift(18,-16)*q^^shift(18,-12)*q, black);
label("A", (6,19), SE);
label("B", (23,4), NW);
label("C", (23,8), NW);
draw((26,11.5)--(30,11.5), Arrows(5));
draw((28,9.5)--(28,13.5), Arrows(5));
label("N", (28,13.5), N);
label("W", (26,11.5), W);
label("E", (30,11.5), E);
label("S", (28,9.5), S);[/asy]