Two probabilities

Con­sider the fol­low­ing state­ments:

1. The re­sult of this coin flip is heads.

2. There is life on Mars.

3. The mil­lionth digit of pi is odd.

What is the prob­a­bil­ity of each state­ment?

A fre­quen­tist might say, “P1 = 0.5. P2 is ei­ther ep­silon or 1-ep­silon, we don’t know which. P3 is ei­ther 0 or 1, we don’t know which.”

A Bayesian might re­ply, “P1 = P2 = P3 = 0.5. By the way, there’s no such thing as a prob­a­bil­ity of ex­actly 0 or 1.”

Which is right? As with many such long-un­re­solved de­bates, the prob­lem is that two differ­ent con­cepts are be­ing la­beled with the word ‘prob­a­bil­ity’. Let’s sep­a­rate them and re­place P with:

F = the frac­tion of pos­si­ble wor­lds in which a state­ment is true. F can be ex­actly 0 or 1.

B = the Bayesian prob­a­bil­ity that a state­ment is true. B can­not be ex­actly 0 or 1.

Clearly there must be a re­la­tion­ship be­tween the two con­cepts, or the con­fu­sion wouldn’t have arisen in the first place, and there is: apart from both obey­ing var­i­ous laws of prob­a­bil­ity, in the case where we know F but don’t know which world we are in, B = F. That’s what’s go­ing on in case 1. In the other cases, we know F != 0.5, but our ig­no­rance of its ac­tual value makes it rea­son­able to as­sign B = 0.5.

When does the differ­ence mat­ter?

Sup­pose I offer to bet my $200 the mil­lionth digit of pi is odd, ver­sus your $100 that it’s even. With B3 = 0.5, that looks like a good bet from your view­point. But you also know F3 = ei­ther 0 or 1. You can also in­fer that I wouldn’t have offered that bet un­less I knew F3 = 1, from which in­fer­ence you are likely to up­date your B3 to more than 23, and de­cline.

On a larger scale, sup­pose we search Mars thor­oughly enough to be con­fi­dent there is no life there. Now we know F2 = ep­silon. Our Bayesian es­ti­mate of the prob­a­bil­ity of life on Europa will also de­cline to­ward 0.

Once we un­der­stand F and B are differ­ent func­tions, there is no con­tra­dic­tion.