The way to look at homeowners insurance is based on an old rule of thumb: When betting against an adversary with deep enough pockets, everybody eventually goes bust. In the case of homeowners insurance, the trusty Interwebz suggests that it costs an average of $800/year in my state. What it means is that as a matter of fact your house is likely to receive less than $800 worth of damage in any year. But self-insurance only works if you can actually afford to cover a big loss, because if you can't, then you have to shut down. Most people in the US can't even cover the loss of a couple of paychecks. For those who could write a check to replace their house, most in that category would still experience a major lifestyle change, especially in retirement.
Likewise in betting, even if the averages are in your favor, eventually you will be killed by a big enough loss, so you can't bet as much as someone who has infinitely deep pockets. The averages are the same for both players, but the rare big events affect big and small players differently. In a betting game the big players will win over time for this reason, even if they can't stack the odds in their own favor.
Still, a bike is not a house. Heck, I could insure my bike by hoarding a spare bike. If you can't discipline yourself to save $50/mo, or don't have the money, then in either case it really means that you can't afford to have that kind of bike.