Independence and probability

Thesis: If we stick to real-numbered probabilities, genuine independence of events A and B cannot be defined in terms of any condition on the conditional probabilities P(X|Y) where X and Y are events can be constructed from A and B by using the boolean operations of union, intersection and complement, even if the conditional probabilities are taken to be primitive rather than defined as ratios.

Argument: Suppose that three genuinely independent darts are thrown uniformly at the interval [0, 1], and consider the events:

• A: the first dart hits [0, 1/2) or the third hits 1/2

• B: the second dart hits [0, 1/2) or the third hits 1/2.

The events A and B are not genuinely independent. The third dart’s hitting 1/2 would guarantee that both events A and B happen. But it is easy to check that the conditional probabilities for any boolean combination of A and B are exactly the same as for the corresponding boolean combination of A′ and B′, where:

• A′: the first dart hits [0, 1/2)

• B′: the second dart hits [0, 1/2).

So, conditional probabilities can’t distinguish the non-genuinely independent pair A and B from the genuinely independent pair A′ and B′.

Nor should we mind this fact. For genuine independence is a concept about causation or rationality, while probabilities give us a statistical concept.