How does Gandalf get informed of Bilbo's 111st birthday party?

Question

In the movie The Lord of the Rings: The Fellowship of the Ring, Frodo Baggins welcomes Gandalf the Grey by remarking:

You're late!

Which implies that he received some form of invitation to Bilbo's eleventy-first birthday party. However, it's also implied several times that he hasn't been in contact with the hobbits in quite a while. This is even more reinforced when Gandalf tells Bilbo that he "hasn't aged a day".

How did Gandalf know that a party was happening or how was he invited? Did he simply remember Bilbo's birthday and realize that it was time to visit again?

Answer 1

According to the book, Gandalf knew everything a very long time ago.

This is implied, even stated, in the very first chapter. Emphasis mine:

Days passed and The Day drew nearer. An odd-looking waggon laden with odd-looking packages rolled into Hobbiton one evening and toiled up the Hill to Bag End.
[...]
When [Gandalf], helped by Bilbo and some dwarves, had finished unloading [...] he disappeared inside with Bilbo, and the door was shut.
[...]
(Gandalf) "You mean to go on with your plan then?"
(Bilbo) "I do. I made up my mind months ago, and I haven’t changed it."
"Very well. It is no good saying any more. Stick to your plan – your whole plan, mind – and I hope it will turn out for the best, for you, and for all of us."
"I hope so. Anyway I mean to enjoy myself on Thursday, and have my little joke."
"Who will laugh, I wonder?" said Gandalf, shaking his head.
"We shall see," said Bilbo.
_{The lord of the Rings, A long expected party}

Mention of "The Day" is of course the 111^th birthday.
Before this paragraph, local folks discuss the oddities of Bilbo, having strange visitors like Dwarves, Gandalf and other outsiders, ever since his return of his adventures.
Bilbo wanted his 111^th birthday to be exceptionnal, way before he reached that age and regardless of how he intended to "disappear".

I guess this is yet another difference when comparing movie / book: hard to squeeze all the details in such a short time.

Answer 2

Not clear

It's left ambiguous by the film whether Gandalf was invited, or whether he merely showed up; the only clue from the dialogue is in an early scene of An Unexpected Journey:

Frodo: You think he'll come?

Bilbo: Who?

Frodo: Gandalf.

Bilbo: Ahhh. He wouldn't miss a chance to lit up his whiz­poppers! He'll give us quite a show, you'll see.

Frodo: Alright then, I'm off.

Bilbo: Off to where?

Frodo: East Farthing woods. I'm going to surprise him.

^{The Hobbit: AN Unexpected Journey (2012)}

From this dialogue, we can gather that:

• He's expected to appear
• They haven't been in contact with him to confirm it

So either they invited him and he didn't reply, or else he wasn't invited but they expected him to show up all the same.

Personally, I find the second of those to be the more likely; considering that the only way to send a message in Middle-earth is by courier¹, and considering that hobbits as a rule don't much like to travel farther than Bree, and further considering that Gandalf is of no fixed address, it seems much simpler to assume that he had merely noted the date and turned up, fireworks in tow.

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¹ The postal system being rather limited to the Shire