Mihai Nica - Alice HH vs Bob HT

A video popped up in my feed this week. It discusses a game which really piqued my attention. See, I actually submitted a very similar game to young students for MATh.en.JEANS this year.

The MATh.en.JEANS game #

The game was the following:

Émilie and Manon play “heads or tails” (H or T) but instead of betting on H or T, they each bet on a pattern. Émilie bets on HTH and Manon bets on HTT. Every time one of their a pattern appears (with overlap), the player who chose it gets a point. The game ends when a player reaches 3 points.

For instance, denoting Émilie’s pattern in italics and Manon’s pattern in bold, here’s a random game:

HTHTTHTTTHHHTT

Here the game stops because Manon wins. After simulating a million games, though, it seems that Émilie wins far more often. How can we explain this?

The point is not for me to give solutions to this problem, so I haven’t thought about the problem too much, but the students found it fun and challenging. They weren’t able to crack it, even after a while. As for my part, poking holes in their conjectures (e.g. “Émilie can overlap, not Manon”) was interesting.

The HH-HT game #

In the video, the initial idea is the same: each player chooses a pattern and counts them with overlap. Except the patterns are shorter and the number of rounds is fixed: the points are counted after 100 flips. Who wins more between HH and HT?

Mihai Nica takes his time to prove that HT wins more, by how much, and how the number of rounds may affect the result. While, the explanation is a bit technical for high-schoolers, the visualisations are great and I found it super engaging! His code is available in a Google Colab in the description, as well.

Any ideas how this can serve as a stepping stone to compare HTT with HTH?