When does a bet fail to reveal your true beliefs? When it hedges a risk in your portfolio.
If this claim does not immediately strike you as obviously true, you may benefit from reading this post by econblogger Noah Smith. Excerpt:
...Alex Tabarrok famously declared that "a bet is a tax on bullshit".
But this idea, attractive as it is, is not quite true. The reason is something that I've decided to call the Fundamental Error of Risk. It's a mistake that most people make (myself often included!), and that an intro finance class spends months correcting. The mistake is looking at the risk and return of single assets instead of total portfolios. Basically, the risk of an asset - which includes a bet! - is based mainly on how that asset relates to other assets in your portfolio.