Play 2*t rounds of the same coin-tossing game, please express P(t rounds show head and the other t rounds show tail, and at any time point between 0 and 2t, the number of coin landing head is no less than that of tail). If possible, please also show that when t goes to infinity, this probability becomes/does not become 0. Please give details even if this is too simple to you This event can also be framed as a random walk: starting at the origin of a vertical axis, in each step, go up (+) and down (-) by 1, never reaching any place that has a negative coordinate (never go downer than the origin), and goes back to origin in the long run. If this probability goes to zero at infinity, please advise on similar events that happen with a positive probability when t goes to infinity.
The probability you seek is $C_t/4^t$, where $C_t$ is the $t$th Catalan number. From the exact formula for $C_t$ you should be able to prove that this probability tends to $0$.