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


If you've talked with me for long enough, you know that 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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