1982 AMC 12 Problem 25

Question

The adjoining figure is a map of part of a city: the small rectangles are blocks and the spaces in between are streets. Each morning a student walks from intersection

A

to intersection

B,

always walking along streets shown, always going east or south. For variety, at each intersection where he has a choice, he chooses with probability

\frac12

(independent of all other choices) whether to go east or south. Find the probability that, on any given morning, he walks through intersection

C.

\frac{11}{32}

\frac12

\frac47

\frac{21}{32}

\frac34

Answer: D

Concepts:basic probability lattice paths binomial probability

Difficulty rating: 2090

Small hint:

Reaching

C

means making the third eastward move before the fourth southward move

Big hint:

Condition on the number

i=0,1,2,3

of south moves made before the third east move

Solution:

If

i

south moves occur before the third east move, the final step to

C

is east and the preceding

2+i

steps contain two east moves. Thus

\begin{aligned} \Pr(C) &=\sum_{i=0}^3\binom{2+i}{2}2^{-(3+i)}\\ &=\frac18+\frac3{16}+\frac6{32}\\ &\quad+\frac{10}{64}\\ &=\frac{21}{32}. \end{aligned}

Therefore, the correct answer is D.