In a cafe I frequent, the area where I sit has a left seat and a right seat, and I enjoy switching where I sit. If I'm in the left seat and I feel like switching, there's a 50% chance that I'll move to the right seat. Similarly, if I'm in the right seat and I feel like switching, there's a 70% chance that I'll switch to the left seat. After much time has gone by, what fraction of time will I have spent sitting in the left seat?
I tried to solve this problem by first letting L be the fraction of time I spent sitting in the left seat, and R be the fraction of time I spent sitting in the right seat. I noticed that $L+R=1$. Then, I let $$L' = 0.5L + 0.5R$$ where L' represents the fraction of time you spend in the left seat in the next iteration. Since $L'=L$ in the long run, the equation becomes $$L = 0.5L + 0.5R$$ I substituted $L+R=1$ in to get: $$L=0.5L+0.5(1-L)$$ $$L=0.5L+0.5-0.5L$$ $$L=0.5$$ Therefore, the fraction of time sitting in the left seat in the long run is 50%. Is this the correct way to approach this problem? Thank you!