Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

A drunkard is 9 steps away from home, and 1 step behind him is a pit. He will move 1 step forward/backward with probabilities of 0.7/0.3 respectively.What are the probabilities that

(a) he will fall into the pit

(b) reach home

(c) reach 1 step away from home

I had solved a somewhat similar problem, but with no pit and a condition of reaching home in 20 steps or less, by analysing patterns on a 2D lattice, getting the formula

sum of (3/n)*C(2n, n+3)*0.7^(n+3)*0.3^(n-3) for n = 3 to 10 = 0.814401

but i don't know how to tackle this question.

I'm not very savvy mathematically, so the simpler the explanation, the more welcome it will be !

share|cite|improve this question
You can read this page on 1-dimensional random walks – utdiscant Apr 25 '12 at 10:19
I have read upto equation 9, but am unable to figure out how the table can be changed when p is not 1/2 – true blue anil Apr 25 '12 at 16:24
up vote 3 down vote accepted

Call $P(n)$ the probability of reaching home if you are $n$ step from the pit. So $P(0)=0$, $P(10)=1$. You are looking for $P(1)$.

For $0<n<10$ $$P(n)=0.7*P(n+1)+0.3*P(n-1)$$ The general solution to this equation is $P(n)=a + b*(0.3/0.7)^n$ (see this link), and using the know value we can write: $P(0)=a+b*1=0$ and $P(10)=a+b*(0.3/0.7)^{10}=1$, and find $$P(n)=a+b*(0.3/0.7)^1=\frac{(0.3/0.7)^n-1}{(0.3/0.7)^{10}-1}$$ $$P(1)=\frac{(0.3/0.7)^1-1}{(0.3/0.7)^{10}-1}$$

The probability of reaching home is $\approx0.5715$ ($P(1)$), the probability of falling $1-P(1)$, and the probability of reaching the last step but don't reaching home is $\frac{(0.3/0.7)^1-1}{(0.3/0.7)^{9}-1}-\frac{(0.3/0.7)^1-1}{(0.3/0.7)^{10}-1}$ because P(reaching the last step AND don't go home) = P(reaching the last step) - P(reaching the last step AND go home) = P(reaching the last step) - P(go home).

share|cite|improve this answer
Thanks a lot for a simple & lucid answer ! – true blue anil Apr 26 '12 at 10:00

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.