The number of paths into the pool at a specified length is given by the sequence \[ C_n = \{1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, \dots \} \]
From OEIS, we find that this sequence is known as the Catalan numbers.
For a brief proof that the sequence of integers generated by Bartschi and Martin (2020) is the Catalan numbers (as well as for proof of the closed form expression of this problem, see the concise proof from Wikipedia Contributors (2019b) here.
From these resources, we learn that the closed form expression of the \(n^\text{th}\) Catalan number is \[ C_n = \binom{2n}{n} - \binom{2n}{n-1} = \frac{1}{n+1}\binom{2n}{n} = \frac{2n!}{n!(n+1)!} \] And therefore the probability of falling in at the \(n^\text{th}\) opportunity would be \[ F_n = \frac{2n!}{n!(n+1)!}\cdot p\big(p(1-p)\big)^n. \]
Finally, using Kolmogorov's 3rd axiom, the probability of the union of all ways of falling in gives the following series: \begin{align*} \mathbb{P}(\text{Falling in} | p) &= \sum_{n=0}^{\infty} F_n \\ &= \sum_{n=0}^{\infty} \frac{2n!}{n!(n+1)!}\cdot p\big(p(1-p)\big)^n \end{align*} where \(n=0\) corresponds with falling in on the first step and \(p\) is the probability of taking a step toward the pool.
Included below is a bit more supplemental information on the Catalan numbers and resources you can use to understand them more fully:
- Gasparovici (2020) notes that the Catalan numbers are ”[p]robably the longest entry inthe Online Encyclopaedia [sic] of Integer Sequences, and rightly so.” The Catalan numbers are named after Belgian mathematician Eugéne Charles Catalan who found the standard formulas in 1838. See this Medium article on Catalan Numbers. Specifically, note we are engaging in the Lattice Paths problem by finding the number of unique paths the Robot can take into the pool.
- Wikipedia Article on Catalan Numbers.
- Another good resource is this mathforum article which contains more pictures but perhaps less explanation.
