When will the haunt begin? – Probably Overthinking It

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One of the favorite board games at my house is Betrayal at House on the Hill.

A unique feature of the game is the dice, which yield three possible outcomes, 0, 1, or 2, with equal probability. When you add them up, you get some unusual probability distributions.

There are two phases of the game: During the first phase, players explore a haunted house, drawing cards and collecting items they will need during the second phase, called “The Haunt”, which is when the players battle monsters and (usually) each other.

So when does the haunt begin? It depends on the dice. Each time a player draws an “omen” card, they have to make a “haunt roll”: they roll six dice and add them up; if the total is less than the number of omen cards that have been drawn, the haunt begins.

For example, suppose four omen cards have been drawn. A player draws a fifth omen card and then rolls six dice. If the total is less than 5, the haunt begins. Otherwise the first phase continues.

Last time I played this game, I was thinking about the probabilities involved in this process. For example:

  1. What is the probability of starting the haunt after the first omen card?
  2. What is the probability of drawing at least 4 omen cards before the haunt?
  3. What is the average number of omen cards before the haunt?

My answers to these questions are in this notebook, which you can run on Colab.

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