Here is a case (something like a real-world case) where an agent accepts things that she regards as having lower probability than things she does not accept. There are a few authors who regard this as impossible or irrational, but this case seems to be perfectly possible, and perfectly rational.
In the story, ESPN is running a viewer competition. One contestant will be chosen as the ‘winner’. If there is an “unassisted triple play”:http://www.baseball-almanac.com/feats/feats8.shtml in any major league baseball game in the month of September, the winner will get $1,000,000. Otherwise they’ll get some token prize.
In most cases where competitions like this are run, the TV station likes to insure against the possibility of someone winning. Susie has been asked to offer a quote on insurance against a victory. Here’s how Susie comes up with her quote, knowing that she doesn’t want to quote too low (because then she’ll be exposing her company to uncompensated risks) or too high (because then the potentially profitable contract will go somewhere else).
Here’s how Susie comes up with her quote. She first works out the historical frequency of unassisted triple plays (utps) in games. Call this _f_. I think _f_ is around 1 in 15,000. Having looked at the unassisted triple plays that happened, and how they related to how the game is played, she concludes that the probability _p_ of an utp in any particular game is equal to _f_, at 1 in 15000.
She then looks up the schedule to work out how many games are left in the season. There are, let’s say, 375 games scheduled for September. Ignoring the chance that there are two utps in the month, she concludes there is about a 1 in 40 chance of there being a utp in the month.
She then adds in a profit margin, and a margin of erros, and comes up with a quote of $35,000.
That seems like a reasonable quote to me, and a reasonable process of generating it. But note one thing Susie ignores. (We’ll assume she consciously ignores this, so this is not ignorance on her part.)
Baseball games in September are often cancelled because of rain and not rescheduled. It is just about a certainty that not all the scheduled games will be played. Indeed, for _any_ game there is a higher probability it will not be played than that there will be a utp in the game. But for each game Susie accepts, while drawing up her quote, that it will be played. That’s why she multiplies by 375 rather than some other number. She does not, on the other hand, accept that for each game there will be no utp in it. If she did, she would assume the contract was pure profit, and make some ridiculous quote like $5000, reducing her company’s expected profits.
The point is that what is reasonable to accept turns on practical consequences rather than just degrees of belief. If Susie were writing insurance for teams against the cost of lost games, she would _not_ accept that 375 games will be played. Indeed, what an agent typically _does_ accept turns on practical consequences not just degrees of belief. I want to say the same is true of belief, which should be much more controversial. But here’s at least a case where acceptance and probability come apart.