Touch-anytime versus at-expiry digital at 30

Interest rates are zero and a stock trades at $20$. Contract A pays $1$ if the stock goes above $30$ at any time during the next year; contract B pays $1$ only if the stock is above $30$ at the end of the year. Compare their values.

A is worth more: a little over twice B. Any path finishing above $30$ must have crossed it, so A dominates B. By the reflection principle, a path that has reached $30$ is about equally likely to end above or below it; exactly, A $=N(d_2)+\tfrac{S}{H}N(d_1)$ with $H=30$ while B $=N(d_2)$, and since a driftless lognormal stock is slightly more likely to end below $30$ after touching it, A exceeds $2B$.