The strategy of the Plinketto board

In RedLetterMedia, they make a type of video called Best of the Worst, where they choose 3 bad movies and watch them, and decide which one of them is the best of the worst. The selection procedure uses a device called a Plinketto board.

I was wondering, what are the likely outcomes from this method?

It is a form of Galton board that is slightly longer. (It has 10 starting slots and length 19). In the Galton board, the results are normally distributed around the center. Thus you can highly influence the results by choosing where you start the drop. The Plinketto board is slightly longer than a Galton board, so the results are not exactly normally distributed when you drop in the center, but it’s close. I performed a million drop simulations using Julia (code).

However, if you drop it near the edge, it looks different:

It is less likely for the ball to end up in the corner spot. Intuitively this is caused by there being two immediate paths to all the other slots, but only one immediate paths to the corner slot. If we look at the distribution of outcomes with random starting slot, it looks like this:

The simulations show that this is the same pattern we get if we use a very long Plinketto board, and drop in the middle. This makes sense, the long drop has the effect of making the starting position essentially random. (Although with lower chance of starting in the edges.) Note that this pattern is only present if the final row is the wider one. If it had been the more narrow one, all slots would have equal chance of being hit, given random drop position.

So, if you are dropping the ball, how large a chance can you get to hit a specific movie? That depends on what slot you aim for.

The method for dropping is to set the ball directly above the movie you are interested in, except if you are interested in one of the movies second or third to the edge, then you drop the ball in the edge slot instead.

In conclusion, when using the Plinketto board:

  1. If you are the movie placer, put the movies you most want to avoid on the edge slots.
  2. If you are both the placer and the dropper, put the movie you are interested in on slot second to the edge, and drop the ball on the edge. This gives you 32% chance to hit your movie.
  3. If you are the dropper, and the movie you want is not second or third from the edge, simply drop the ball above it. This gives you at least 17.6% chance to hit your movie.
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