Subjective probability

by Paul Crowley

Credits: This way of looking at probability is due to Bruno de Finetti; this particular framing was taught to me by Andrew Critch.

Out of the blue, you get the following email from me:

Dear You:

I extend to you, and you alone, a chance to take part in my free lottery. Please choose at most one of the following options:

  • Option A: On November 9th, I’ll roll two standard six-sided dice, and if I roll a double one (“snake-eyes”), I’ll send you $200
  • Option B: On November 9th, if Washington DC has been declared for Clinton, I’ll send you $200

If I don’t hear from you within 24 hours, or if your answer isn’t a clear preference for one of these, I won’t do either. Thanks!

If you know me, you know that I’d straightforwardly honour the promise in the email; for this thought experiment set aside all questions about that. It may help you to know that Obama got over 90% of the vote in DC in 2012. There seems zero benefit in refusing the offer or not replying – it’s totally free, and the worst that can happen if you lose is that I don’t send you $200.  Would you reply to this email, choosing one of the options? If so, which one? I think it’s obvious what the right choice is, but stop for a moment and decide what you’d do.

What about if I’d offered you the opposite choice: A’ means $200 if I don’t roll snake-eyes, while B’ means $200 if Clinton doesn’t win DC? Does that change your choice?

I hope you chose B in the first case, and A’ in the second. That’s because it seems very clear that Clinton’s chances of winning in DC are very high, and certainly higher than the chances of snake-eyes on a single roll of two dice, which is 1/36 or less than 3%.

However, for many people, this statement contradicts their understanding of what probability is. The most common and widely taught view of probability is strictly frequentist; it makes sense to say that a pair of dice have a less than 3% chance of snake-eyes only because you can roll the dice many times and in the long run they will land snake-eyes one time in 36. You cannot rerun the 2016 Presidential election in DC many times, so it means nothing to say that Clinton has a greater than 3% chance of winning.

If you’re prepared to choose between the options above – if you agree that a single roll of a pair of dice producing snake-eyes is less likely than a Democratic victory in DC in 2016 – then there’s an important sense in which you already reject this view and accept a subjective view of probability.