Odds of winning the lottery: a distance analogy

We don’t have a good intuition for long odds, so I started playing around with an analogy for the odds of winning the lottery.

Imagine a line 9 miles long. Have you ever run for 9 miles? It gets boring. You could traverse a reasonably-sized city, like Manchester, or get from Nottingham to Derby. Properly imagine it: running, at a decent pace, through towns and villages and countryside, for one and a half hours straight.

Now imagine I’ve chosen a random point along the way, and to win the lottery you have to stop there. Maybe you stop after one step, or half a step, or at the Gallows Inn in Ilkeston, or just past the road that goes to Dale Abbey, or outside the Co-op in Spondon. To win the lottery, you’d have to stop at the right point to the millimetre.

Or in terms of time: throughout the one and a half hours, you’d be travelling at about 2,680 millimetres per second. That means stopping at a time resolution of something like the half-millisecond, which is sixty times higher than the framerate of our vision.

Naturally, I wanted to expand the analogy to account for the fact that sometimes people do win, which connects to the inverse bias – a lack of intuition for large numbers. If 30 million people ran from Nottingham to Derby at once, maybe one or two of them would hit the mark. But to fit that many people, the track would have to be 6,750 miles wide: the runners at the north side of the track would actually be in the Greenland Sea, and those at the south side would be somewhere off the coast of Namibia.