But if someone did in the first few weeks I'd be very suspicious of the honesty of the lottery.
Only because you (like someone who knows how probability works) are comparing the number of trials (whatever percent of the population buys lottery tickets) with the sample space (all the possible lottery tickets). That's a far cry from an argument from personal incredulity ("I just can't believe that, so it must not be true.")
We make decisions and take actions on probablities when they are 19 to 1 or 99 to 1 in "favor". When we do this (say in medical treatment tests) we are treating (tentively) .05 or .01 as being equal to zero.
Sure, in the case of one thing happening once, that's probably good enough. But in the case of one improbable thing being tested over and over again, over time, the odds of any improbable thing happening rise dramatically. If you have infinite time, all improbable things occur.
Now, of course, we don't have infinite time. The question is really "what improbable things could happen during the lifetime of the universe?" Luckily abiogenesis appears to be one of those things.
But ultimately, I agree with you - probability isn't even a cogent argument here because we simply don't know what factors were involved in abiogenesis, so we can't set up any kind of probability after the fact.