Solution to May's Puzzle: Missing Element in Series

I got this nice puzzle from Adi Dafni. Given the following series of elements, can you determine the next element?

1, 11, 21, 1211, 111221, 312211, ?

Each element describes the previous element by counting the number of consecutive occurrences of each digit in order. For example, to describe the element 1, you would say that there's one occurrence of the digit 1, resulting in 11. To describe 11, you would say that there are two occurrences of the digit 1, resulting in 21. The description of 21 is one occurrence of 2 and one occurrence of 1, resulting in 1211. The description of 1211 is one occurrence of 1, one occurrence of 2, and two occurrences of 1, resulting in 111221. Following this logic, the next few elements are 312211, 13112221, and 1112213211.

June's Puzzle: Same Birthday

What's the probability that, in a group of 23 randomly chosen people, at least two of them will have the same birthday?