Thursday, January 1, 2009

Let's start with a little probability intuition

Imagine you're playing a board game where your pieces move according to a die roll. Unless it's a very dull game, there will probably be other things that can influence your position as well, but we're ignoring them right now.

Let's imagine that normally your move is given by a single die roll (six sided die, d6), and you pass along the spaces only once - it's a straight "race", not a circuit.

No doubt you've seen many games where you move along spaces to some end, and a few spaces have special characteristics (such as "miss a turn" or "move ahead three spaces"). BoardGameGeek calls these "roll and move" games, and appears to list more than six thousand of them. If that doesn't ring a bell, think of something a bit like Snakes and Ladders.
(However, you can apply similar ideas to a large variety of games where you accumulate some resource with varying probabilities. )

Further, let's say there's a particularly desirable space to land on (which I'll call the target). You're currently 6 spaces away (that is, if you roll a 6 you will land on the space). You really would like to land on that space, either this turn, or next turn, or the turn after that, ... &c.

Quick - without working it out exactly, and assuming for now that no strategy is available to change the probability, what would you guess would roughly be your chance of hitting the the target at all?

Then, if you're so inclined, try to work out the probability. (If you're not inclined to attempt it, never fear - I will post the correct answer later. There's at least one fairly easy way to do it - and a lot of somewhat less easy ways to do it.)

This was one case where my intuition wasn't as directly useful as I might hope. Intuition served me just fine for figuring out what the probability would be from far away, but from closer up, not so much - at least not at first.

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