You really shouldn't apply statistics this way. Expected value is useful if you're an insurance company because you can price insurance products slightly above the expected value, but if the fine is $100, batteries are $90, and you get fined after going on 10 bike rides the fine doesn't magically become $10 it's still $100 and you still could've saved $10 by spending $90 before getting on your bike. That is, unless you put $10 in a piggy bank every time you get on your bicycle, and then at that point why not put $90 down and then pay yourself back $9 per ride with $1 of interest?

I mean that if fine / E[N_f] < battery_cost / E[N_b], where E[N_] is an expected number of rides before a fine / battery expire, then I save money per ride. In your example, you just pick the values such that the relation becomes '>'.