I need help in the following question:
Two robots are located at X=0 on the X-axis. Every turn each robot moves a unit forward or backward in equal probability 0.5, in a manner that is independent between the robots and between the different turns. Let D be the distance between the robots after 450 turns. Calculate approximatly P(D$\leq$6).
Answer by the book: 0.158.
I am almost certain that by "approximatly" they want to treat n=450 as infinite and use the CLT. Using indicators I got to P(-0.013$\leq$Z$\leq$0.013) (where Z is normal) but it seems to be wrong... I can find the final number from the distribution table myself, I only need to know what did I do wrong.
Thanks in advance
Some hints: A given robot's position after 450 steps will be $X_1+X_2+\cdots+X_{450}$ where each $X_i$ has mean zero and variance $1$ (why?). So by the CLT the position of robot A is approximately normally distributed, with mean $0$ and variance $450$ (why?). The position of robot B has the same distribution, and is independent of robot A's position.
So you are looking for $P(|X-Y|\le 6)$, where $X$ and $Y$ are independent, and normally distributed with mean $0$ and variance $450$. You will recognize that the difference $X-Y$ has a normal distribution, since it's the difference of two independent normal variables; what are the mean and variance of $X-Y$? This should be enough to lead to a solution.