Daily plus or minus fifty percent over a year

Each trading day a trader's capital either rises by $50\%$ (probability $0.6$) or falls by $50\%$ (probability $0.4$), independently. (a) What is the probability that after $260$ trading days he has more than he started with? (b) Over which number of days is the probability of being ahead largest?

(a) About $0.14$. (b) Three days, with probability $0.648$. After $n$ days with $k$ up-days wealth is $3^k/2^n$, so he is ahead iff $k>n\ln2/\ln3\approx0.631n$; for $n=260$ that needs $k\ge165$, and the binomial tail is $\approx0.141$. Expected log growth is negative, so the chance decays with horizon; checking small $n$, $n=3$ (needing $2$ or more ups) is best.