I am not a lawyer, but this is how these licenses are usually interpreted in my experience:
The MIT license is a so-called permissive license. It explicitely allows merging and sublicensing. That means you can use MIT-licensed code and relicense it under a different license. This allows you to use the code in any project regardless of the license.
The only condition you have to fulfill is that you need to include the copyright notice and permission notice of the MIT-licensed component in your project. You don't present this notices in a way that it looks as if it applies to your whole project. You should just make clear that your product includes MIT-licensed code, who made it and what licenses apply to the code you used. Common practice is to put these in the readme-file of a non-gui application or the about-section of a GUI application.
The Microsoft Public License is a lot more convoluted than MIT, but similar in spirit. It is also a permissive license which also allows to merge and sublicense the resulting product. The MS-PL puts an additional restriction for sublicensing: "If you distribute any portion of the software in compiled or object code form, you may only do so under a license that complies with this license". Whether or not a license complies with the MS-PL is a topic you can debate about (The GNU-GPL does not, but I am quite sure the MIT does), but in your case it's irrelevant. You intend to license the resulting software under the MS-PL. That means it does not concern you, because the MS-PL code retains its license.