You have two coins in front of you: red and blue. The blue coin flips a head with probability $\frac{1}{8}$ and the red coin flips a head with probability $\frac{1}{9}$.
• If both coins flip heads, you go north $3$ blocks.
• If blue is heads and red is tails, you go north $1$ block.
• If blue is tails and red is heads, you go east $1$ block.
• If both coins flip tails, you go east $2$ blocks.
What is your expected location after $217$ flips? What is the standard deviation for going north in exactly $1$ flip of both coins? Assume coin flips are independent for each coin and that both coins are independent. You must use random variables.
Calculating the individual probabilities for the $4$ directions is pretty straightforward. The issue I am facing is the non-homogenous nature of the random walk. And secondly, how do I go about the $217$ flips. One thing that comes to my mind is using complex numbers, mapping the directions in $2D$ complex plane and each subsequent flip would simply be the multiplication of the difference in coordinates. I am not sure if this would work. Looking for any more leads and/or developments on my idea.
Thanks.
Let $X_1$ be the random variable that represents the first step of the walk.
It is equal to $3e_2$ with probability $1/72$, $e_2$ with probability $1/9$, $e_1$ with probability $7/72$, and $2e_1$ with probability $7/9$, where $e_2$ is north and $e_1$ is east.
Let $X_1, X_2, \ldots, X_{217}$ be iid. We want to know $\mathbb{E}[X_1 + X_2 + \cdots + X_{217}]$. Due to the linearity of expectation, this is equal to $$\mathbb{E}[X_1] + \mathbb{E}[X_2]+\cdots+\mathbb{E}[X_{217}] = 217\mathbb{E}[X_1].$$
Can you take it from here?