# Expected Number of Distinct Subsequences in Randomly Generated Binary Strings

## TL;DR

This paper investigates the expected number of distinct subsequences in randomly generated binary strings, generalizing known results to include biased probabilities, arbitrary alphabets, and Markov chain models.

## Contribution

It extends existing formulas for expected distinct subsequences to biased binary strings, arbitrary alphabets, and Markov chain-generated strings.

## Key findings

- Derived formulas for biased binary strings with probability α
- Extended analysis to arbitrary alphabet sizes
- Analyzed Markov chain string generation models

## Abstract

When considering binary strings, it's natural to wonder how many distinct subsequences might exist in a given string. Given that there is an existing algorithm which provides a straightforward way to compute the number of distinct subsequences in a fixed string, we might next be interested in the expected number of distinct subsequences in random strings. This expected value is already known for random binary strings where each letter in the string is, independently, equally likely to be a 1 or a 0. We generalize this result to random strings where the letter 1 appears independently with probability $\alpha \in [0,1]$. Also, we make some progress in the case of random strings from an arbitrary alphabet as well as when the string is generated by a two-state Markov chain.