Or more generally, X sends a costly signal of his belief in P. If X is the state (as in example 2) a bet is probably impractical, but doing anything that would be costly if X is false should work.
But for this, it makes a big difference in what sense Y does not trust X. If Y thinks X may deceive, costly signals are good. If Y thinks X is stupid or irrational or similar, showing belief in P is useless.
Or more generally, X sends a costly signal of his belief in P. If X is the state (as in example 2) a bet is probably impractical, but doing anything that would be costly if X is false should work. But for this, it makes a big difference in what sense Y does not trust X. If Y thinks X may deceive, costly signals are good. If Y thinks X is stupid or irrational or similar, showing belief in P is useless.