Drawing the Same Item More Than Once is Surprisingly Likely

When drawing items from a pool with replacement, the probability that you’ll draw the same item more than once is much higher than intuition would suggest.

Often called the Birthday Paradox and framed thus: in a group of 23 people, there is a greater than 50% chance that at least two people will share a birthday.

Math

Let $k$ be the number of items we draw with replacement from the pool with $n$ items. $p(n,k)$ is the probability that at least one item is drawn more than once.

p(n,d)=\begin{cases} 1 - \frac{_{n}P_{k}}{n^{k}} &k \leq n \\ 1 &k \gt n \end{cases}

In SageMath:

def birthday(n, k):
    if k > n:
        return 1
    return 1 - (Permutations(n, k).cardinality() / (n ^ k))

Notes Linking Here

Learning Sample Rules Without Replacement to Avoid Repetitive Procgen