Dice displays to prompt intuitive understanding of probabilities

About a week and a half ago, when I looked, iPredict showed approximately a 63% probability that there would be a National Prime Minister after the 2017 general election in New Zealand, and a 37% probability that there would be a Labour Prime Minister. Here’s what that looks like as a horizontal bar:bar

The problem with this way of looking at it is that (especially in New Zealand, with a proportional voting system), there’s a temptation to interpret the proportions as vote shares, rather than as probabilities. And even when I’ve got the idea of vote shares out of my mind, I can still be inclined to interpret it as a prediction that National will win the 2017 election. It’s not; it’s an estimate that there is a 63% probability that National will win the election.

How can I encourage myself to understand this intuitively?

Well, here’s my idea: Superimpose the image of the faces of a six-sided die on the bar I showed above, like this:example-1 Now I can imagine a game in which a six-sided die is thrown: if it shows a number less than 3, then Labour wins; if it shows a number greater than 3, National wins; if it shows exactly 3, then either might win, with the probability proportional to the relative areas shown on that face of the die.

(I don’t mean to suggest that this is how elections are actually decided in New Zealand; this is just a way of giving myself a good intuition about the probabilities involved.)

But this isn’t entirely satisfactory yet. What happens in that imaginary game to decide the winner if a 3 is shown? Well, I can superimpose an image of the faces of a second die on that face of the first one, like this:example-2 If a 3 is shown on the first die, a second is thrown: if it shows less than 2, Labour wins; if it shows greater than 2, National wins; if it shows exactly 2, then the game is still undecided.

In principle, the superimposition of the faces of more dice could be continued as far as you like, using the base-six (or senary, or perhaps, in this case, dicemal) expansion of the probability in question. But I’m not sure it’s worth it; beyond two dicemal places, the the faces of the dice would be hard to see without using a very zoomable image, and it probably wouldn’t add much to the intuition about the probability.

There might be one exception to that last comment, though: If the less probable event is very unlikely, you could dispense with the image, and just say that it is, for example, “less likely than throwing five dice and having them all show sixes”. Then, anyone sufficiently familiar with Yahtzee would understand that it’s particularly unlikely, but not entirely impossible.

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