A stock today price 100, tomorrow it becomes 50 or 200. There is also a casino you could bet win/loss with rate 1:1. What is your strategy to make money?
I denote the probability of stock rises the other day as p and betting it goes down with money x. Then the expectation of gain is $f(p)=p(100-x)+(1-p)(-50+x)=(1-2p)x+150p-50$
When $0<p<\frac{1}{2}$,$f(p)>0$ is $x>-75-\frac{25}{2p-1}$
When $\frac{1}{2}<p<1$, I got $x<-75-\frac{25}{2p-1}$
Can’t find an arbitrary? What’s wrong?