Learn French: Est-ce que tu as un billet ?

It acts like the English word "Do"! Literally, est-ce que means "is it that". You can put it in front of any normal statement to turn it into a Yes/No question. Since tu as un billet means "you have a ticket", adding Est-ce que at the start makes it "Do you have a ticket?".

Why doesn't que shrink to qu' before tu?

In French, you only drop the final e in que when the very next word starts with a vowel or a silent h. Since tu starts with a consonant (t), que stays whole. If the subject were il (he), you would drop the e and write Est-ce qu'il.

I've heard the word ticket in French. Why use billet here?

Both mean "ticket", but they are used for different things. Un billet is typically for trains, airplanes, or larger events like concerts (it also means a bank note). Un ticket is usually for local transport like a city bus or metro, or a printed receipt from a cash register.

Do I have to use Est-ce que to ask this question?