If it was not possible to download in order, things like popcorn wouldn't be there for years. It streams torrent movies via this method.

No, it can download files in order. Almost all clients have that option.

Isn't the random order option better in some way? Is there a way of using probability to quantify this in some way? I'm thinking that there's a higher probability that a random "chunk" of a file will be available if the chunks are downloaded randomly. But I haven't looked into this enough to know.

I'd assume you are correct (intuitively, if you don't constrain order at all, you'll have more opportunities for matching required parts than if you do constrain order) However I've found that it's very much possible to stream 1080p torrents without any problems, even when skipping back and forth or jumping to other parts of the movie without anymore noticeable delay than I'd expect from e.g. netflix.