Imagine if you had a precise prior that made you certain of \(X\) (this is compatible with the above constraints since \(X\) was unconstrained). Seeing \(X\) land uppermost now should be evidence that the coin has landed heads. The game set-up makes it such that these apparently irrelevant instances of evidence can carry information. Likewise, being very confident of \(X\) makes \(Y\) very good evidence for \(H\). If instead you were sure \(X\) was false, \(Y\) would be solid gold evidence of \(H\)’s falsity. So it seems that \(p(H\mid Y)\) is proportional to prior belief in \(X\) (indeed, this can be proven rather easily). Given the way the events are related, observing whether \(X\) or \(\neg X\) landed uppermost is a noisy channel to learn about whether or not \(H\) landed uppermost.