Interpretation of conditional probability

What does it mean? Textbooks present it as a definition (a kind of 4th axiom), although very often, a few lines later in the same book, the formula is presented as a theorem (!).

In the subjective approach, one is allowed to talk about independently of and . In fact, is just the assessment of the probability of , under the condition that is true. Then it cannot depend on the probability of . It is easy to show with an example that this point of view is rather natural, whilst that of considering () as a definition is artificial. Let us take

- = Higgs mass of 250 GeV;
- = the decay products which are detected in a LHC detector;
- the evaluation of is a standard PhD student task. He chooses GeV in the Monte Carlo and counts how many events pass the cuts (for the interpretation of this operation, see the previous section). No one would think that must be evaluated only from and , as the definition () would imply. Moreover, the procedure is legitimate even if we knew with certainty that the Higgs mass was below 200 GeV and, therefore, .

In the subjective approach, () is a true theorem required by coherence. It means that although one can speak of each of the three probabilities independently of the others, once two of them have been elicited, the third is constrained. It is interesting to demonstrate the theorem to show that it has nothing to do with the kind of heuristic derivation of Section :

- Let us imagine a coherent bet
on the conditional event to win a unitary
amount of money (, as the scale factor is inessential).
Remembering the meaning of conditional probability in terms of bets (see
Section ),
this means that
- we pay (with certainty) ;
- we win 1 if and are both verified (with probability );
- we get our money back (i.e. ) if does not happen (with probability ).

- The expected value of the `gain'
is given by the probability of each
event multiplied by the gain associated with each event:

E

where the first factors of the products on the right-hand side of the formula stand for probability, the second for the amount of money. It follows that

- Coherence requires the rational better to be indifferent to the direction of the bet, i.e. E. Applying this condition to () we obtain ().