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.

• 0.62
• 0.64
• 0.66
• 0.68
• 0.7

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.