A random walk is a stochastic process that moves in discrete steps randomly. In this simple random walk, after each step, the process has equal probability of moving up or down by 1 unit.
The probability of hitting a point in a simple symmetric random walk before another can be calculated based on the distances from the start to each point. Let’s use this to solve your problem:
Let’s denote the points as A = +3 and B = -2.
The total distance between the two points is |A - B| = |3 - (-2)| = 5 units.
The distance from the starting point (0) to B is 2 units.
In a symmetric random walk, the probability of hitting point A before B is equivalent to the distance from the starting point to B divided by the total distance, as this represents the proportion of the total walk that must be covered to reach B. So:
P(A before B) = Distance from start to B / Total distance = 2 / 5 = 0.4 or 40%.
So, the probability of hitting +3 before hitting -2 in a simple random walk with an equal probability of moving up or down, starting from 0, is 40%. This seems more intuitive, as +3 is further away from the starting point than -2.