There are already multiple billions of people who have access to Spotify right now. Spotify is available on iOS, Android, Windows, Mac and the Web. That's a huge potential market.
Yet Spotify only has 20 million paid customers... and 55 million free users. That's kind of a low percentage, isn't it?
So will 1 billion Apple users give Apple Music 100 million users? I don't think we can predict that.
I've been questioning how big the streaming market is in general. There are obviously lots of people and devices on Earth... but that doesn't mean they'll sign up for your service (or pay for it!)
There are some critical differences here though. Even-though Spotify is available on Android, iOS and Windows Phone (the Windows Phone app is still rather bad..) as well as in your browser, desktop, various other devices etc.. Apple has one big advantage, their solution will be on every single iOS device upgraded to iOS 8.4 and beyond.
You don't need to know about Apple Music beforehand, you don't need to head over to the AppStore, search for Spotify, install it, create your account to get everything going. Most of Apple's iOS users already got their Apple ID with their credit card ready to go. All they need to do is hit the "Music" button and I bet the first thing you will be seeing is a "Hey! Would you like to try our awesome unlimited Apple Music service? It's free for three months!" and all you need to do is to push the button, verify with your Apple ID password or use Touch ID and you are good to go.
I'm certain that the "Tips" application and various things in the new "pro-active" Siri search when you swipe to the leftmost home page in iOS 9 will have links to Apple Music all over it, so it will be pretty much impossible to ignore it.
The thing is, Spotify is far from being equally accessible at least on iOS devices. And you also have to remember, quite a few people still use iTunes for Music. They might not purchase that many songs or albums any more but tons of people still have a large library of music in iTunes already that they keep accessing. I guess the new Apple Music desktop application is going to replace iTunes all-together and even those who do not update will most likely see references to Apple Music all over the iTunes Store within iTunes. And again, all these people already have their Apple ID logged in, registered with their credit card ready to roll.
Not to mention.. They offer a freaking three months free trial. THREE MONTHS... I bet a bunch of existing Spotify users, even users that don't plan on leaving Spotify will be tempted to simply pause their subscription and try Apple Music for three months and there's a big chance that they won't be going back as long as Apple is able to deliver a good application, a good service and a large library of music.