Oh, and then there are the times when you’ll give different probabilities for the same event, if the question is worded in different ways.
That implies a very easy Dutch-book:
Create a lottery with three possible outcomes (a), (b), and (c) - for example, (a) 1, (b) 2, 3, or 4, and (c) 5 or 6 on a six-sided die. (Note that the probabilities are not equal—I have no need of that stipulation.)
Ask “what are the odds that (a) will happen?” In response to the proposed even-odds, bet against (a).
Ask “what are the odds that (b) will happen?” In response to the proposed even-odds, bet against (b).
Ask “what are the odds that (c) will happen?” In response to the proposed even-odds, bet against (c).
Collect on two bets out of three, regardless of outcome.
That implies a very easy Dutch-book:
Create a lottery with three possible outcomes (a), (b), and (c) - for example, (a) 1, (b) 2, 3, or 4, and (c) 5 or 6 on a six-sided die. (Note that the probabilities are not equal—I have no need of that stipulation.)
Ask “what are the odds that (a) will happen?” In response to the proposed even-odds, bet against (a).
Ask “what are the odds that (b) will happen?” In response to the proposed even-odds, bet against (b).
Ask “what are the odds that (c) will happen?” In response to the proposed even-odds, bet against (c).
Collect on two bets out of three, regardless of outcome.