48. A Brief Note On My Weird Choices of Probability Figures


If you've talked with me for long enough, you know that I cleave to that ancient and honored tradition of putting numbers to my probability estimates. Be it guesses at whether my flight will be on time, how likely I think I am to get accepted for a job or grant, or how confident I am in my prediction of someone's taste in candy, I'll happily pause for a moment or three, do a little Focusing on numbers, poke my mental model, and spit out a number.

The weird thing about my estimates is, they don't tend to be round tens of percents. Often, they aren't even simple ratios of small whole numbers - 3s and 93s and 35s abound. So what's up with that? Am I just posing? Wonderfully, no! I like to think about probabilities in terms of successive coin flips - that is, something like surprisal, or log-probability. So when I spout an "87", I'm thinking that it'd be as surprising as seeing three successive heads rather than anything else - meaningfully odd, but not out of the question.

Then why the 35s? I noticed a few years back when I began once more to think about probabilities and predictions with explicit numbers attached that to use just whole numbers of flips is extremely limiting. Most notably, you have a giant gap between 25 and 75 broken only by the "I have no idea" of 50. To a lesser extent, the same problem occurs between 12~13 and 25, and 75 and 87~88. To me, the solution was clear - use half-flips as well! I don't have a particular favorite physical interpretation of what it means for something to be 1/2√2 ~ 35% likely (a flip and a half coming up your way) but I permit myself the sloppiness of mostly rounding off mentally to 1/3 ~ 33%; likewise for 1/4√2 ~ 18% and 1/6 ~ 17%.

In the interests of completeness, I've provided a fair-sized table below of approximate probability, 1 - probability, "flips' worth of surprise", and how I think about the subjective experience of what it's like to hit that chance of 1 in whatever:

50        50        1 flip        as likely as not; not surprising

35        65        1½ flip        a little unexpected; noteworthy

25        75        2 flips         a little surprising; "oh huh, guess it's that today"

18        82         2½ flips        surprising; not something to count at all on, like a 6 on a d6

12        87        3 flips        moderately surprising; kind of unexpected but not outside imagination

6        93        4 flips        very surprising; wasn't really planning for this

3        97        5 flips        straight-up shocking; less than a crit-success

1.6       98.4        6 flips         extremely surprising; was probably counting on not-this

0.8        99.2        7 flips        straight-up blatantly suspicious; confusing

0.1        99.9        10 flips        what? no. no!; straining credulity

(increasingly absurdly unlikely/a-sure-thing; I don't generally bother with anything smaller than 1/128)

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