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Which method performs better: .Any() vs .Count() > 0?

I just wonder why should I use Any() instead of Count()?, if we took the msdn example :

class Pet
{
    public string Name { get; set; }
    public int Age { get; set; }
}
class Person
{
    public string LastName { get; set; }
    public Pet[] Pets { get; set; }
}

public static void AnyEx2()
{
    List<Person> people = new List<Person>
        { new Person { LastName = "Haas",
                       Pets = new Pet[] { new Pet { Name="Barley", Age=10 },
                                          new Pet { Name="Boots", Age=14 },
                                          new Pet { Name="Whiskers", Age=6 }}},
          new Person { LastName = "Fakhouri",
                       Pets = new Pet[] { new Pet { Name = "Snowball", Age = 1}}},
          new Person { LastName = "Antebi",
                       Pets = new Pet[] { }},
          new Person { LastName = "Philips",
                       Pets = new Pet[] { new Pet { Name = "Sweetie", Age = 2},
                                          new Pet { Name = "Rover", Age = 13}} }
        };

    // Determine which people have a non-empty Pet array.
    IEnumerable<string> names = from person in people
                            where person.Pets.AsQueryable().Any()
                            select person.LastName;

    foreach (string name in names)
        Console.WriteLine(name);

    /* This code produces the following output:

       Haas
       Fakhouri
       Philips
    */
}

What if I used :

  IEnumerable<string> names = from person in people
                            where pers
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