LINQ with inner enumerable query

I'm struggling to get my head around LINQ. In the example below I have three meetings. Jane Doe is assigned to two of those meetings. But the code below which I think should produce an answer of two is showing three meetings. Any ideas?

class Program
{
    static void Main(string[] args)
    {
        Person p1 = new Person() { Name = "John Doe" };
        Person p2 = new Person() { Name = "Jane Doe" };
        Person p3 = new Person() { Name = "Alan Doe" };
        Meeting m1 = new Meeting() { MeetingName = "Meeting One", People = new List<Person> { p1, p2 } };
        Meeting m2 = new Meeting() { MeetingName = "Meeting Two", People = new List<Person> { p2, p3 } };
        Meeting m3 = new Meeting() { MeetingName = "Meeting Three", People = new List<Person> { p1, p3 } };
        List<Meeting> allMeetings = new List<Meeting>{m1,m2,m3};
        //Now, find all the meetings that Jane Doe is in
        var results = allMeetings.Select(x => x.People.Select(y => y.Name == "Jane Doe").Any());
        Console.WriteLine(results.Count().ToString());
        Console.ReadLine();
    }
}
public class Meeting
{
    public string MeetingName { get; set; }
    public List<Person> People { get; set; }
}
public class Person
{
    public string Name{get;set;}
}