You can think of insurance as a small & certain loss to hedge against a large & uncertain loss.
If stuff happens and you do need to cash in on that insurance policy, the payout should be thought of as saving your butt, also known as indemnification.
An airbag in your car is a form of insurance. You spent $X to protect yourself in a collision. The small & certain loss is spending the $X. Say you unexpectedly end up in a crash, but instead of dying, the airbag saves your life. It protected you from a death, the large & uncertain loss.
The idea of that buying an insurance is making a bet against yourself doesn't make sense to me. Insurance is more about making sure you don't lose. It's reducing risk, and a bet feels like taking on risk. The insurance company is the one making the bet, not the policy holder.