Two drunks start out together at the origin, each having...

Two drunks start out together at the origin, each having equal probability of making a step to the left or right along the $x$ axis. Find the probability that they meet again after $N$ steps. It iz to be understood that the men make their steps simultaneously. (It may be helpful to consider their relative motion.)

Key Concepts

Binomial Distribution

The binomial distribution describes the probabilities of obtaining a certain number of successes (or moves in one direction) out of a fixed number of independent trials. In the context of random walks, it is used to compute the likelihood of being at a particular position after a given number of steps, forming the basis for combinatorial analyses of meeting probabilities.

Relative Motion

Considering the relative motion between two random walks simplifies the problem of determining when and where the walks meet. By focusing on the difference between their positions, the problem is reduced to analyzing a single random walk that represents the displacement between the two. The meeting event corresponds to this displacement returning to zero.

Random Walk

A random walk is a mathematical model in which an object makes a series of moves or steps, each determined by chance. In one dimension, this often means taking steps to the left or right with certain probabilities. This model is foundational in probability theory and is used to represent various real-world processes characterized by randomness.

Independent Random Processes

When two random walks occur independently, each process evolves without influencing the other. This independence allows for the analysis of combined systems, such as comparing the positions of two walkers, by considering their individual probability distributions and later combining the results to study intersections or meeting events.

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