Queue for five-dollar tickets with no change

Tickets cost $5. A queue of $2n$ people forms in random order: $n$ of them pay with a $5 bill and $n$ with a $10 bill. The box office opens with no cash. What is the probability that the cashier always has a $5 bill available whenever someone pays with a $10?

$\tfrac{1}{n+1}$. Treat a $5 payer as a step up and a $10 payer as a step down; the walk must never go below zero. The number of such paths is the Catalan number $\tfrac{1}{n+1}\binom{2n}{n}$, out of $\binom{2n}{n}$ equally likely orderings.