A random walker starts at 0 on the [math]x[/math]-axis and at each time unit moves 1 step to the right or 1 step to the left with probability 1/2. Estimate the probability that, after 100 steps, the walker is more than 10 steps from the starting position.

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Solution: D

We let [math]X_i = \pm 1 [/math] with probability that [math]X_i = 1 [/math] equals 1/2. Then we want to approximate the probability that [math]\sum_{i=1}^{100}X_i \notin [-10,10] [/math]. The expected value of [math]X_i [/math] is 0 and its variance is 1. By the central limit theorem, the sum [math]\sum_{i=1}^{100}X_i [/math] is approximately normally distributed with mean 0 and variance 100. Hence we have

[[math]]\sum_{i=1}^{100}X_i \notin [-10,10] \approx P(Z \notin [-1,1])[[/math]]

where [math]Z[/math] is a standard normal. This is approximately equal to 0.3173.