I just tried to install Android Studio on my Windows 10 system. When I open Android Studio it ask me where my Android SDK is. But according to the documentation http://developer.android.com/sdk/installing/index.html?pkg=studio
Android Studio provides everything you need to start developing apps for Android, including the Android Studio IDE and the Android SDK tools.
So what's gives? I guess Android Studio does NOT provide everything ...
You are correct that it does not provide everything as stated in the documentation elsewhere. Android Studio is an IDE - Integrated Development Environment - designed for developing Android apps in particular. Android SDK - Software Development Kit - is a group of tools for building an app that targets a specific Android API.
In other words, the Android SDK's are specific to developing an app based on target set of Android tools, components, etc. Therefore, in Android Studio, you can setup one or more Android SDK's for developing many apps. This includes custom SDK's like from Samsung, Honeywell, Zebra or other Android device manufacturers.
In Android Studio, you can select which of these SDK's you would like to install and use. If you do not select any SDK, Android Studio will be useless. It also should not include a specific SDK since you may not want or need it. Also, SDK's get updated independent of Android Studio, so you need to manage that separately.
As CloudBrain explains, you can setup an Android SDK in the same directory as other Android tools. But you may not, and probably should not, do that in order to keep that directory clear for Android Studio to manage. Selecting any available directory should work - and then download and install your preferred SDK(s) or use Android Studio to do that.
The link at the bottom of the page that you reference is here:
http://developer.android.com/sdk/installing/adding-packages.html
And it states:
Adding SDK Packages
By default, the Android SDK does not include everything you need to start developing. The SDK separates tools, platforms, and other components into packages you can download as needed using the Android SDK Manager. So before you can start, there are a few packages you should add to your Android SDK.
In other words, the SDK itself comes kind of "empty" and you need to define how you will use it by adding the components/tools/platform versions/etc. that you will need for developing your particular app. To include all of this would severely bloat the download and installation.