How Do Random Walks Affect a Tourist's Position in New York City?

[joestats]

I am completely lost on how to solve this problem. Any hint would help

There is a lost tourist in New York. The streets in New York run east to west and go from

..., S. 2nd St., S. 1st St., Broadway St., N. 1st St., N. 2nd St., ...
The avenues run north to south and go from

..., E. 2nd Ave., E. 1st Ave., Broadway Ave., W. 1st Ave., W. 2nd Ave., ...
These streets form a square block grid. For each of the questions below, the tourist starts at the intersection of Broadway St. and Broadway Avenue and moves one block in each of the four cardinal directions with equal probability.

What is the probability that the tourist is at least 3 city blocks (as the crow flies) from Broadway and Broadway after 10 moves?

What is the probability that the tourist is at least 10 city blocks (as the crow flies) from Broadway and Broadway after 60 moves?

What is the probability that the tourist is ever at least 5 city blocks (as the crow flies) from Broadway and Broadway within 10 moves?

What is the probability that the tourist is ever at least 10 city blocks (as the crow flies) from Broadway and Broadway within 60 moves?

What is the probability that the tourist is ever east of East 1st Avenue but ends up west of West 1st Avenue in 10 moves?

What is the probability that the tourist is ever east of East 1st Avenue but ends up west of West 1st Avenue in 30 moves?

What is the average number of moves until the first time the tourist is at least 10 city blocks (as the crow flies) from Broadway and Broadway.

What is the average number of moves until the first time the tourist is at least 60 city blocks (as the crow flies) from Broadway and Broadway.

 

[Taryn1]

Have you tried drawing the problem out and visualizing the roads? It can really help, for a start.

 

[joestats]

Yes I did.. But I am new to probability. I just need an example somewhere that I can look and then tackle this problem

 

[MaryM]

Re: Path Problem with Probability. Help Please

Yes I did.. But I am new to probability. I just need an example somewhere that I can look and then tackle this problem

See Ross A First Course in Probabiiity Chapter 1 eighth edition

Mary

 

[HOI]

"What is the probability that the tourist is at least 3 city blocks (as the crow flies) from Broadway and Broadway after 10 moves?"
is a simple example since only involves 10 moves. "3 city blocks (as the crow flies)" would be a circle with center at "Broadway and Broadway" and radius 3: \(\displaystyle x^2+ y^2= 9\) where x is the net number of blocks moved east or west and y is the net number of blocks moved north or south. The opposite of "at least 3 city blocks" is "fewer than 3 city blocks" so it is simpler to calculate the probability that the tourist is fewer than 3 city blocks, then subtract that from 1.