2 Probabilism

2.1 Assigning Probabilities#

We now turn to assigning probabilities to events, and to what the constraints on doing so are. In particular, we’re interested in the structure of credence: the constraints placed by epistemic rationality on how our credences should look.

2.2 Epistemic irrationality#

Intuitively, some patterns of credence we can have are incorrect.

Have a think about each of these, and about what is wrong with having credences like this.

Example
Joe thinks that there is a 50% chance that it will rain today and a 20% chance that it will not rain today.
Example
Biqing thinks that there is a 60% chance she failed the exam she just took, and a 70% chance she failed it badly.
Example
Homer thinks that there’s a 30% chance he’ll drink one beer, a 50% chance he’ll drink more than one beer, and a 70% chance he’ll drink at least one beer.
2.3 The Probability Axioms (again)#

Lecture 1 stated these informally:

  • (i) Non-negativity. Probabilities can’t be negative.
  • (ii) Normalisation. The probability of the universal event—of something happening—is 1.
  • (iii) Additivity. The probability of either of two completely separate events happening is the sum of their probabilities. (E.g., the probability of a rolled die landing 1 or 2 is \nicefrac{1}{6} + \nicefrac{1}{6} = \nicefrac{1}{3}.)

We now have names for all the pieces. \Omega is the sample space, \Sigma is the set of events, and \Cr(\sigma) is the credence assigned to an event \sigma \in \Sigma. “Completely separate” is disjointness, which we defined above as X \cap Y = \varnothing.

  • (i) Non-negativity: \Cr(\sigma) \geq 0 for every \sigma \in \Sigma.
  • (ii) Normalisation: \Cr(\Omega) = 1.
  • (iii) Finite additivity: if X \cap Y = \varnothing, then \Cr(X \cup Y) = \Cr(X) + \Cr(Y).
  • More generally, if the events \sigma_0, \sigma_1, \dots are pairwise disjoint, then \Cr(\bigcup_i \sigma_i) = \Cr(\sigma_0) + \Cr(\sigma_1) + \dots
2.4 So what went wrong?#

We can now say exactly which axiom each of the three cases above breaks.

  • Joe — normalisation, via additivity. Rain and no-rain are disjoint and between them exhaust \Omega, so by additivity their credences must sum to \Cr(\Omega), which normalisation fixes at 1. His sum to 0.7.
  • Biqing — additivity, together with non-negativity. Failing badly is a way of failing, so failing badly is a subset of failing, and failing splits into failing badly and failing-but-not-badly. Additivity makes \Cr(\text{fail}) the sum of those two, and non-negativity keeps the second one from being negative; so failing badly cannot be more probable than failing. Hers is 0.7 against 0.6.
  • Homer — finite additivity. One beer and more than one beer are disjoint, and together they are “at least one beer”. So that credence should be 0.3 + 0.5 = 0.8. His is 0.7.