Reversal invariance of expected occurrence time

Prove that reversing any finite binary ending string S preserves its expected first-occurrence time, namely, establish E(S)=E(S') whenever S' is the reversal of S.

Background

The paper establishes reversal-type equalities for several families of ending strings, including two-run, three-run, and four-run strings. These results motivate a broader symmetry claim concerning arbitrary finite strings of heads and tails.

The conjecture asserts that the expected waiting time for a string and for the string read in reverse are identical. The authors explicitly state that they do not have a rigorous proof, offering only an intuitive argument based on average distances between successive occurrences when a long sequence is read forward or backward.

References

This leads to another conjecture. Conjecture 4.2. If S' is the reversal of an ending string S, then E(S) = E(S'). We do not have a rigorous proof for the above conjecture, but here is an intuitive argument: if one flips a coin a large number of times, then E(S) is the average distance between an occurrence of S and the next, but reading the outcomes backward, we have the same average distance between an occurrence of S' and the next, so E(S') = E(S).

— A coin flip game and generalizations of Fibonacci numbers  (2501.07463 - Huang, 13 Jan 2025) in Conjecture 4.2, Section 4