# Chance of random walk returning to origin

1. Dec 9, 2008

### Loren Booda

Starting out at zero on a number line and moving in succession one unit right or left at random, what is the probability that you will eventually return to zero?

2. Dec 9, 2008

3. Dec 9, 2008

### Loren Booda

I guess that, in terms of limits, it approaches 1.

4. Dec 9, 2008

Well, probably 1 has a precise meaning, and that isn't that it is certain to happen. See almost surely on Wikipedia.

You are correct that the probability approaches one as you allow yourself to take more steps on the random walk.

5. Dec 10, 2008

### noisysignal

yup , 1 it is

6. Dec 10, 2008

### CRGreathouse

Assuming moving in each direction is done with probability 0.5, the probability is indeed 1. If the probability of moving left is different, the probability drops below 1.

7. Dec 10, 2008

### Staff: Mentor

Note that it doesn't mean that if probability moving right is different, the probability raises above 1.

More seriously, if we are allowed to walk for ever probability of visiting any point approaches 1.

8. Dec 10, 2008

### CRGreathouse

I don't think so. If you move left with 60% probability, I calculate the probability of return as between 71.57% and 71.58%. (Once you start going left, you risk never coming back.) You have a finite expected maximum excursion to the right in that case.

9. Dec 10, 2008

### Staff: Mentor

I meant when both directions are 50/50.

10. Dec 12, 2008

### kenewbie

Any ideas on how to compute the chance of an outcome in an infinite series of weighted coin-tosses?

ie, if my coin has a probability of .7 to land heads, what is the propability of an infinite string of tosses that are all heads?

k

11. Dec 12, 2008

### Staff: Mentor

What is probability of H? HH? HHH?

12. Dec 12, 2008

### CRGreathouse

If the probability of landing heads is less than 1, the probability of an infinite string of heads is 0. I'm not sure how to show this, though, since you don't require the string to start with the first flip (so no finite portion is sufficient to reject).

13. Dec 12, 2008

### kenewbie

Sure, it approaches zero, I can agree with that.

I dont know how to do calculate a series yet (let alone infinite ones) but I have trouble wrapping my head around the IDEA of this. There must be some chance, however small? The argument that sooner or later it has to hit heads just smells funny. Rationally I see that it will hit zero if I try it, but if one imagine an infinite amount of infinite strings, then I cant see that there isn't room for one in which all are heads. In fact, there should be room for an infinite amount of all-heads strings.

k

14. Dec 13, 2008

### CRGreathouse

I said that the probability was 0, not that it was impossible. Search Google (or the search engine of your choice) for "almost certain".

Yes, there are an infinite number of all-heads strings (one for each zero-terminated finite string, plus one for the null string, all followed by infinitely many 1s). But this is a countable infinity in an uncountable ocean of endless strings. (If "countably infinite" and "uncountably infinite" are unknown to you, either search for this or ignore this point.)

15. Dec 14, 2008

### rscosa

Hi!
the answer of this problem is not known yet. I know the answer if you are in a line at some point and you wanna know the probability to return to 0 which can be written in terms of the hyperbolic sine by using really complicated mathematics (not for undergraduates).